<PAGE>   1
                       SECURITIES AND EXCHANGE COMMISSION
                              Washington D.C. 20549



                                    FORM 6-K



                        REPORT OF FOREIGN PRIVATE ISSUER
                    PURSUANT TO RULE 13a-16 OR 15d-16 OF THE
                         SECURITIES EXCHANGE ACT OF 1934


                  For the quarterly period ended March 31, 1998


                          ----------------------------



                                   OLICOM A/S
                 (Translation of registrant's name into English)


                                  Nybrovej 114
                                 DK-2800 Lyngby
                                     Denmark
                    (Address of principal executive offices)


                          -----------------------------


       [Indicate by check mark whether the registrant files or will file
              annual reports under cover of Form 20-F or Form 40-F

                           Form 20-F [X] Form 40-F [ ]

        [Indicate by check mark whether the registrant by finishing the
       information contained in this Form is also thereby furnishing the
         information to the Commission pursuant to Rule 13g3-2(b) under
                      the Securities Exchange Act of 1934.

                                 Yes [ ] No [X]

         [If "Yes" is marked, indicate below the file number assigned to
        the registrant in connection with Rule 13g3-2(b): Not Applicable

             This Form 6-K shall be incorporated by reference to the
           registrant's registration statement on Form F-4 under the
                Securities Act of 1933, as amended, registration
                                 no. 333-24655.


<PAGE>   2



                                   OLICOM A/S
                                    FORM 6-K

                                TABLE OF CONTENTS


<TABLE>
<CAPTION>
                                                                                          Page
                                                                                          ----

<S>       <C>                                                                            <C>
ITEM 1.   FINANCIAL INFORMATION

          Consolidated Balance Sheets as of
             December 31, 1997 and March 31, 1998                                            3

          Consolidated Statements of Income
             for the three months ended March 31, 1997
             and March 31, 1998                                                              4

          Consolidated Statements of Cash Flows for the three months
             ended March 31, 1997 and March 31, 1998                                         5

          Notes to Consolidated Financial Statements                                         6


ITEM 2.   MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION
             AND RESULTS OF OPERATIONS                                                       8


ITEM 4.   EXHIBITS

Exhibit No.                         Description
-----------                         -----------

     99.1                   Press Release dated April 22, 1998, relating                     12
                            to Financial Results for First Quarter 1998
</TABLE>



                                  Page 2 of 14



<PAGE>   3


                                   OLICOM A/S

                           CONSOLIDATED BALANCE SHEETS

<TABLE>
<CAPTION>
                                                        DECEMBER 31, 1997          MARCH 31, 1998
                                                        -----------------          --------------
                                                                                     (UNAUDITED)
                                                                      (IN THOUSANDS)
<S>                                                        <C>                       <C>      
ASSETS

Current assets:
       Cash and cash equivalents                           $  45,591                 $  35,025
       Short term investments                                    915                       945
       Accounts receivable, less allowance of
              $3,326 in 1997 and $2,990  in 1998              57,967                    62,659
       Inventories                                            25,663                    27,869
       Deferred tax                                            1,925                     2,608
       Prepaid expenses and other                              4,337                     7,658
                                                           ---------                 ---------
              Total current assets                           136,398                   136,764
Property and equipment                                        16,028                    17,475
Purchased intangibles                                          9,172                     8,672
Investment in affiliated companies                               733                       844
                                                           ---------                 ---------

              Total assets                                 $ 162,331                 $ 163,755
                                                           =========                 =========

LIABILITIES AND SHAREHOLDERS' EQUITY
Current liabilities:
       Accounts payable                                    $  18,815                 $  16,243
       Accrued payroll and related expenses                    4,092                     4,356
       Accrued product warranty expense                        1,118                     1,118
       Other accrued expenses                                  8,834                     9,397
       Deferred revenue                                        2,942                     2,676
       Income taxes payable                                       45                        13
                                                           ---------                 ---------
              Total current liabilities                       35,846                    33,803


Minority interests                                               926                       788
Shareholders' equity:
       Common shares, DKK 0.25 nominal value
              authorized and issued -
              18,495 in 1997, 18,495 in 1998                     712                       712
       Additional paid - in capital                          102,633                   102,681
       Retained earnings                                      38,728                    40,094
       Treasury stock - 940 in 1997 and 874 in 1998          (14,988)                  (13,092)
       Unearned compensation                                  (1,594)                   (1,329)
       Unrealized gains/losses on securities                      68                        98
                                                           ---------                 ---------
           Total shareholders' equity                        125,559                   129,164
                                                           ---------                 ---------
              Total liabilities and shareholders' equity   $ 162,331                 $ 163,755
                                                           =========                 =========
</TABLE>



                             See accompanying notes




                                  Page 3 of 14
<PAGE>   4

                                   OLICOM A/S

                        CONSOLIDATED STATEMENTS OF INCOME

<TABLE>
<CAPTION>
                                                                               THREE MONTHS ENDED
                                                                                   MARCH 31,
                                                                          -----------------------------
                                                                              1997            1998
                                                                          -------------   -------------

                                                                     (IN THOUSANDS, EXCEPT PER SHARE AMOUNTS)
                                                                                    (UNAUDITED)
<S>                                                                       <C>              <C>        
Net Sales                                                                 $     48,404    $     65,897
Cost of Sales                                                                   26,149          34,597
                                                                          ------------    ------------

       Gross profit                                                             22,255          31,300
                                                                          ------------    ------------

Operating expenses:
       Sales and marketing                                                      10,486          16,524
       Research and development                                                  2,896           5,656
       General and administrative                                                2,095           3,210
                                                                          ------------    ------------

           Total operating expenses                                             15,477          25,390
                                                                          ------------    ------------

       Income from operations                                                    6,778           5,910
Interest income (expense) and other, net                                           420             564
                                                                          ------------    ------------

Income before income taxes                                                       7,198           6,474
Income taxes                                                                     2,520           1,913
                                                                          ------------    ------------

Income before minority interests                                                 4,678           4,561
Minority interests                                                                  82             (38)
                                                                          ------------    ------------

       Net income                                                         $      4,596    $      4,599
                                                                          ============    ============

Net income per share (diluted EPS)
                                                                          $       0.31    $       0.25
                                                                          ============    ============

Weighted average shares outstanding
Including common stock equivalents                                              14,934          18,627
                                                                          ============    ============

Net income per share (basic EPS)                                          $       0.31   $        0.26
                                                                          ============   =============

Weighted average shares outstanding                                             14,678          17,574
                                                                          ============   =============
</TABLE>





                             See accompanying notes




                                  Page 4 of 14
<PAGE>   5

                                   OLICOM A/S

                      CONSOLIDATED STATEMENTS OF CASH FLOWS

<TABLE>
<CAPTION>
                                                               THREE MONTHS ENDED
                                                                    MARCH 31,
                                                                1997          1998
                                                              -------        ------
                                                                    (UNAUDITED)
                                                                  (IN THOUSANDS)
<S>                                                          <C>           <C>
Operating activities:
    Net income                                                $  4,597      $  4,599
    Adjustments to reconcile net income to net
       cash provided by (used in) operating activities:

       Depreciation and amortization                               981         2,491
       Equity in net income of affiliate                             0          (111)
       Minority interest in earnings                                99           (38)
       Exchange rate adjustment re. minority interest              (17)         (100)
       Deferred income taxes                                       (18)         (683)
                                                                               
       Purchased research and development                            0             0
       Change in operating assets and liabilities:
           Accounts receivable                                   (3101)       (4,692)
           Other receivables                                      (147)            0
           Inventories                                           1,020        (2,206)
           Prepaid expenses                                     (1,125)       (3,322)
           Accounts payable                                     (5,941)          105
           Accrued payroll and related expenses                    131           264
           Accrued product warranty expense                         42             0
           Other accrued liabilities                             5,169        (2,379)
           Income taxes payable                                 (1,285)          (32)
                                                              --------      --------
              Net cash provided by (used in)
                  operating activities                             405        (6,104)
Investing activities:
    Capital expenditures                                        (1,117)       (3,437)
    Proceeds from sale of property and affiliated company            0             0
    Investment in affiliated company Digianswer                      0             0
    Proceeds from sale of short-term investments                     0           265
    Acquisition of CrossComm net of cash acquired                    0             0
    Acquisition of Lasat net of cash acquired                        0             0
                                                              --------      --------
              Net cash provided by (used in)
                  investing activities                          (1,117)       (3,172)
 Financing activities:
    Borrowings (repayments)                                         (0)           (0)
    Proceeds from warrants/options exercised                       197         1,932
    Sale (purchase) of treasury stock                               (0)       (3,221)
                                                              --------      --------
              Net cash used in financing activities                197        (1,289)
Effect of exchange rate changes on cash                           (879)           (0)
                                                              --------      --------
Net increase (decrease) in cash and cash equivalents            (1,394)      (10,565)
Cash and cash equivalents at beginning of period                41,664        45,591
                                                              --------      --------
Cash and cash equivalents at end of period                    $ 40,270      $ 35,026
                                                              ========      ========
Interest paid during the period                               $      4      $     16
                                                              ========      ========
Tax paid during the period                                    $  3,944      $  2,845
                                                              ========      ========
</TABLE>


                             See accompanying notes





                                  Page 5 of 14
<PAGE>   6

                                   OLICOM A/S

                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS

The Consolidated Financial Statements as of March 31, 1997 and 1998, are
unaudited. In the opinion of the management of Olicom A/S (the "Company"), such
unaudited financial statements include only such normally recurring adjustments
necessary for a fair presentation of the results of operations for the interim
periods presented and of the financial position of the Company at the date of
the interim balance sheet. The results for such interim periods are not
necessarily indicative of the results for the entire year.

It is recommended that this financial data be read in conjunction with the
audited consolidated financial statements and notes thereto included in the 1997
Annual Report.

1.  EARNINGS PER SHARE

    In 1997, the Financial Accounting Standards Board issued Statement of
    Financial Accounting Standards No. 128 (SFAS 128). "Earnings Per Share". The
    Standard is effective for both interim and annual periods ending after
    December 15, 1997. The Company has adopted SFAS 128 retroactively.
    Accordingly, the Company has disclosed both Basic earnings per share and
    Diluted earnings per share for all periods presented. Diluted earnings per
    share are computed based on the weighted average number of common stock and
    common stock equivalents outstanding during each year. Common stock
    equivalents are determined under the assumption that outstanding warrants
    and options are exercised. Outstanding warrants and options have been
    included in earnings per share computations based on the treasury stock
    method. Basic earnings per share are computed based on the weighted average
    shares outstanding during each year.

2.  PRINCIPLES OF CONSOLIDATION

    The consolidated financial statements include the accounts of Olicom A/S and
    its majority-owned subsidiaries. The Company's investments in 20-50% owned
    companies are accounted for by the equity method of accounting.

 3. INVENTORIES

    Inventories are stated as the lower of cost or market with cost determined
    on the basis of the first in, first out method. Raw materials inventories
    are sold at the Company's cost to subcontractors who assemble products to
    the Company's specifications. Finished goods inventories include completed
    products purchased from subcontractors. Inventories consist of:

<TABLE>
<CAPTION>
                                                            DECEMBER 31, 1997      MARCH 31, 1998
                                                            -----------------      --------------
                                                                                     (UNAUDITED)
                                                                         (in thousands)

<S>                                                             <C>                  <C>      
    Finished goods                                              $  17,704            $  20,991
    Raw materials                                                   7,959                6,878
                                                                ---------            ---------
       Total inventories                                        $  25,663            $  27,869
                                                                =========            =========
</TABLE>

4.  LEASEHOLD IMPROVEMENTS AND EQUIPMENT

    Leasehold improvements and equipment are carried at cost. Depreciation is
    charged on a straight-line basis to costs and expenses over the expected
    useful lives of the assets. Equipment is depreciated over four years, and
    leasehold improvements are amortized over the shorter of their estimated
    lives or non-cancelable term of the lease. Leasehold improvements and
    equipment consist of:





                                  Page 6 of 14
<PAGE>   7

                                   OLICOM A/S

                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS




<TABLE>
<CAPTION>
                                                              DECEMBER 31, 1997    MARCH 31, 1998
                                                              -----------------    --------------
                                                                                     (UNAUDITED)
                                                                          (in thousands)

<S>                                                                 <C>              <C>      
    Leasehold improvements                                          $   2,970        $   3,171
    Equipment                                                          31,809           34,695
                                                                    ---------        ---------
                                                                       34,779           37,866
    Accumulated depreciation                                          (18,751)         (20,391)
                                                                    ---------        ---------

       Total property and equipment                                 $  16,028        $   17,475
                                                                    =========        ==========
</TABLE>

5.  REPORTING CURRENCY

    Although the Company and its subsidiaries maintain their books and records
    in local currencies, as required by law, the Consolidated Financial
    Statements have been prepared in U.S. dollars because the U.S. dollar is the
    currency of the primary economic environment in which the Company and its
    subsidiaries conduct their operations.

    The majority of the Company's sales are billed and collected in U.S.
    dollars, and the majority of the Company's purchases of raw materials and
    finished goods inventories are invoiced and paid in U.S.
    dollars.

6.  RESEARCH AND DEVELOPMENT COSTS

    Research and development costs, including costs of developing software
    products, are expensed as incurred. Application of Statement of Financial
    Accounting Standards No. 86, "Accounting for the Costs of Computer Software
    to Be Sold, Leased, or Otherwise Marketed", has not had any material effect
    on the Company's consolidated financial position or results of operations.





                                  Page 7 of 14
<PAGE>   8

                                   OLICOM A/S

MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS OF
OPERATIONS FOR THE THREE MONTHS ENDED MARCH 31, 1997 AND 1998.


Certain statements included in this Report include trend analysis and are
forward-looking statements (within the meaning of Section 27A of the Securities
Act of 1933, as amended, and Section 21E of the Securities Exchange Act of 1934,
as amended), including, without limitation, statements containing the words
"believes", "anticipates", "expects" and words of similar import. Such
forward-looking statements relate to future events, the future financial
performance of the Company, and involve known and unknown risks, uncertainties
and other factors that may cause the actual results, performance or achievements
of the Company or industry results to be materially different from any future
results, performance or achievements expressed or implied by such
forward-looking statements. Prospective investors should specifically consider
the various factors identified in this Report that could cause actual results to
differ, including, without limitation, those discussed in the following section,
as well as in the Company's Annual Report on Form 20-F. The Company disclaims
any obligation to update any such factors or to publicly announce the results of
any revisions to any of the forward-looking statements contained herein to
reflect future events or developments. The following discussion should be read
in conjunction with the Consolidated Financial Statements and related notes.

The following discussion should be read in conjunction with the information
contained in the Company's Annual Report on Form 20-F for the fiscal year ended
December 31, 1997, and the consolidated financial statements and related notes
included elsewhere herein.

RESULTS OF OPERATIONS

Net sales increased $17.5 million, or 36.1%, from the three months ended March
31, 1997, to the comparable period of 1998. The Company believes that the
increase in net sales during such period was due to several factors, including
the revenues attributable to the former CrossComm Corporation (CrossComm), which
was acquired on June 12, 1997, increased sales in the U.S. in general, and
continued strength in the Company's traditional lines of business (especially,
growth in sales of Token-Ring PCI adapters). Net sales during the first quarter
of 1998 also benefitted from the introduction in mid-September, 1997 of the
Company's Token-Ring and ATM switch products.

Gross profit increased $9.0 million, or 40.66%, from the three months ended
March 31, 1997, to the comparable period of 1998, and increased as a percentage
of net sales from 46.0% to 47.5%. The increase in gross margins was primarily
due to a favorable change in product mix and the inclusion of revenue from
CrossComm's operations, which typically operates at a higher average gross
margin than the Company experienced during the comparable period in 1997.

Sales and marketing expenses increased $6.0 million, or 57.6%, from the three
months ended March 31, 1997, to the comparable period of 1998, and increased as
a percentage of net sales from 21.7% to 25.1%. The increase in the amount of
such expenses during such period was primarily due to increased marketing
activities both in the United States and Europe, including higher costs
associated with personnel expenses and promotional expenditures and the
inclusion of expenses associated with CrossComm's operations. The Company
expects that increased levels of sales activity will continue to require the
commitment of additional resources to the sales and marketing of the Company's
products. Accordingly, marketing expenses are expected to continue to be a
significant percentage of net sales as a result of the planned continued
investment in the Company's sales and support organization.

Research and development expenses increased $2.8 million, or 95.3%, from the
three months ended March 31, 1997, to the comparable period of 1998, and
increased as a percentage of net sales from 6.0% to 8.6%. The increase primarily
reflects the research and development activities that were acquired as part of
the CrossComm transaction and the hiring of additional personnel required to
support enhancements of current products and expenditures for new product
development. All of the Company's research and development costs have been
expensed as incurred.





                                  Page 8 of 14
<PAGE>   9

General and administrative expenses increased $1.1 million, or 53.2%, from the
three months ended March 31, 1997, to the comparable period of 1998, and
increased as a percentage of net sales from 4.3% to 4.9%. The increase in the
amount of such expenses during such period primarily reflected the inclusion of
expenses associated with CrossComm's operations and additional personnel and
costs associated with the generally higher level of business activity.

The Company's operating income for the first quarter of 1998 was $5.9 million,
compared to operating income of $6.8 million for the first quarter of 1997.

LIQUIDITY AND CAPITAL RESOURCES

The Company has two unsecured line of credit facilities for an aggregate amount
of DKK 15.0 million ($2.1 million) and, in addition, a USD-denominated line of
credit facility for $8.5 million, of which $10.6 million was unused at March 31,
1998. Under prevailing banking practice in Denmark, these lines of credit are
terminable by the lender on 14 days prior notice (even if the Company is not in
breach of the general conditions for such facilities) and are terminable without
notice in the event of a breach thereof by the Company (subject to any
applicable cure period).

On March 31, 1998, the Company's inventory levels had increased 8.6% from that
recorded at December 31. 1997. This increase was primarily due to lower sales
than expected in the first quarter of 1998.

Trade accounts receivable at March 31, 1998, increased 8.1% to $62.7 million,
from that recorded at December 31, 1997. The increase was primarily due to sales
revenue in the first quarter being recognized later than normal.

BUSINESS ENVIRONMENT AND RISK FACTORS

      The Company's future operating results may be affected by various trends
and factors, which the Company must successfully manage in order to achieve
favorable operating results. In addition, there are trends and factors that are
beyond the Company's control that may affect its operations.

      Such trends and factors include, without limitation, the following:
conditions within the networking industry, and economic conditions generally;
rapid technological change, frequent product introductions, changes in customer
needs and evolving industry standards, which require that the Company continue
to add engineering refinements to its existing products and develop and
introduce new products which achieve market acceptance; difficulties or delays
in the development, production and marketing of products, including, without
limitation, any failure to ship new products and technologies when anticipated
and a failure of manufacturing economies to develop when planned; fluctuations
in the Company's revenues and operating results from quarter to quarter, due to
a variety of factors, including, among others, the timing of significant orders,
the timing of product introductions by the Company or its competitors,
variations in net revenues by product and distribution channel, increased price
and other competition, and decisions by distributors and OEMs as to the quantity
of the Company's products to be maintained in inventories; pricing, purchasing,
operational and promotional decisions by distributors, value added resellers and
OEMs, which could affect their supply of, or end-user demand for, the Company's
products; the absence of long-term obligations on the part of distributors and
OEMs to purchase products from the Company (and the implicit risk of any
unanticipated declines in sales to any of the Company's material customers for
competitive reasons or because of the internalization of production of products
purchased from the Company on an OEM basis); the Company's shipment of products
shortly after receipt of a purchase order, with the result that a substantial
portion of the Company's revenues for any quarter results from orders received
during such quarter, and minor shifts in the timing of purchase orders can have
a significant effect on net sales for any quarter; the Company's failure to
accurately anticipate the demand for its products, due to, among other things,
the fact that the Company's expectations of future net sales as well as its
expenditures are based largely on its own estimate of future demand and not on
firm customer orders; declines in the demand for Desktop Network Interface
cards, which accounted for approximately 53.9% of the Company's net sales during
1997; the effect that consolidation in the LAN industry may have on the
competitive position of the Company and its revenues and operating results; the
acquisition of assets and businesses and the making or incurring of any





                                  Page 9 of 14
<PAGE>   10

expenditures and expenses in connection therewith, including, without
limitation, any research and development expenses relating thereto; the ability
of the Company to reduce product and other costs; the activities of any parties
with whom the Company has an agreement or understanding, including, without
limitation, issues affecting joint development projects in which the Company is
a participant; unexpected changes in regulatory requirements, tariffs and other
trade barriers, longer accounts receivable payment cycles and other risks
associated with international operations; the ability or inability of the
Company to hedge against foreign currency, exchange rates and fluctuations in
such rates; and a change in the value of the U.S. dollar (the Company's
functional currency) relative to other currencies.

In light of the foregoing factors, as well as other factors affecting the
Company's operating results, past trends should not be used by investors or
others to anticipate future trends, and prior operating performance may not be
an accurate indicator of future performance.

SAFE HARBOR STATEMENT UNDER THE PRIVATE SECURITIES LITIGATION REFORM ACT OF 1995

With the exception of historical information, certain of the matters discussed
in this report and the press release included herewith are forward-looking
statements that involve risks and uncertainties, including, without limitation,
the risks and uncertainties described above under the caption "Business
Environment and Risk Factors", together with such risks and uncertainties as are
described in registration statements, reports and other documents filed by the
Company from time to time with the Securities and Exchange Commission pursuant
to the Securities Act of 1933, as amended, and the Securities Exchange Act of
1934, as amended. Such risks and uncertainties could cause the Company's actual
consolidated results for 1998 and beyond to differ materially from those
expressed in any forward-looking statements made by, or on behalf of, the
Company.





                                 Page 10 of 14
<PAGE>   11

                                   SIGNATURES


The registrant certifies that it meets all of the requirements for filing and
has duly caused this form to be signed on its behalf by the undersigned,
thereunto duly authorized.



                                                Olicom A/S



Date:   August 5, 1998                          By: /s/ Boje Rinhart
                                                --------------------
                                                    Boje Rinhart
                                                    Chief Financial Officer





                                 Page 11 of 14
<PAGE>   12
                               INDEX TO EXHIBITS



<TABLE>
<CAPTION>
EXHIBIT
NUMBER                             DESCRIPTION
-------                            -----------
<S>               <C>

  99.1            Olicom Reports Revenue and Earnings for First
                  Quarter 1998
</TABLE>