<PAGE>   1


                       SECURITIES AND EXCHANGE COMMISSION
                            WASHINGTON, D.C.  20549
                                --------------

                                   FORM 10-Q

                QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d)
                     OF THE SECURITIES EXCHANGE ACT OF 1934

                 For the quarterly period ended March 31, 1998

                         Commission file number 0-10810

                              KIEWIT ROYALTY TRUST
          ------------------------------------------------------------
             (Exact Name of Registrant as Specified in Its Charter)


<TABLE>
<S>                                                        <C>                                          
            Nebraska                                                       47-6131402                   
----------------------------                               ------------------------------------------   
(State or Other Jurisdiction of                               (I.R.S. Employer Identification No.)      
 Incorporation or Organization)

</TABLE>

                                 Trust Division
                         U.S. Bank National Association
                               1700 Farnam Street
                               Omaha, Nebraska  68102
                      ------------------------------------
                    (Address of Principal Executive Offices)

       Registrant's Telephone Number, Including Area Code: (402) 348-6000



                 Indicate by check X whether the registrant: (1) has filed all
reports required to be filed by Section 13 or 15(d) of the Securities Exchange
Act of 1934 during the preceding 12 months (or for such shorter period that the
registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days    Yes   X    No
                                                -----     -----

                 As of May 8, 1998, there were 12,633,432 Units of Beneficial
Interest of the registrant outstanding.
<PAGE>   2
                                     PART I
                             FINANCIAL INFORMATION

ITEM 1.  FINANCIAL STATEMENTS.

                              KIEWIT ROYALTY TRUST
               STATEMENT OF ASSETS, LIABILITIES AND TRUST CORPUS
                   as of March 31, 1998 and December 31, 1997

<TABLE>
<CAPTION>
ASSETS                                                1998                         1997
------                                                ----                         ----
<S>                                        <C>                          <C>
Cash and short-term investments            $     2,535,543              $       180,832

Royalty and overriding
royalty interests in coal leases           $       167,817              $       167,817

Less accumulated amortization              $       (89,838)             $       (85,611)

Net royalty and overriding
royalty interests in coal leases           $        77,979              $        82,206
                                           ---------------              ---------------

Total Assets                               $     2,613,522              $       263,038
                                           ===============              ===============


LIABILITIES AND TRUST CORPUS
----------------------------

Distributions payable to Unit Holders      $     2,535,543              $       180,832

Trust Corpus:  12,633,432
Units of Beneficial Interest
authorized and outstanding                 $        77,979              $        82,206
                                           ---------------              ---------------

Total Liabilities and Trust Corpus         $     2,613,522              $       263,038
                                           ===============              ===============



</TABLE>

                  The accompanying notes are an integral part
                          of the financial statements.





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<PAGE>   3
                              KIEWIT ROYALTY TRUST

                       STATEMENT OF DISTRIBUTABLE INCOME
               for the Three Months Ended March 31, 1998 and 1997


<TABLE>
<CAPTION>
                                                                Three Months Ended March 31
                                                                ---------------------------

                                                       1998                                           1997
                                                       ----                                           ----
<S>                                        <C>                                              <C>
Royalty Income                             $      2,534,970                                 $     1,850,105

Interest Income                            $         18,955                                 $        16,342

Trust Expenses                             $        (18,382)                                $       (14,888)
                                           -----------------                                 ---------------

Distributable Income                       $      2,535,543                                 $     1,851,559

Distributable Income per Unit
(12,633,432 Units Outstanding)                    0.2007010                                       0.1465603

</TABLE>


                      STATEMENT OF CHANGES IN TRUST CORPUS
               for the Three Months Ended March 31, 1998 and 1997


<TABLE>
<CAPTION>
                                                                Three Months Ended March 31
                                                                ---------------------------

                                                      1998                                             1997
                                                      ----                                             ----
<S>                                        <C>                                              <C>
Trust Corpus as of January 1               $        82,206                                  $        90,896

Amortization of Royalty Interest           $        (4,227)                                 $          (884)

Distributable Income                       $     2,535,543                                  $     1,851,559

Distribution to Unit Holders               $    (2,535,543)                                 $    (1,851,559)
                                           ----------------                                 ---------------- 

Trust Corpus as of March 31                $        77,979                                  $        90,012
                                           ===============                                  =============== 

</TABLE>



                See accompanying Notes to Financial Statements.





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<PAGE>   4
                              KIEWIT ROYALTY TRUST

                         Notes to Financial Statements
                   as of March 31, 1998 and December 31, 1997

Basis of Presentation

                 The accompanying unaudited financial statements have been
prepared in accordance with the instructions for Form 10-Q and do not
necessarily include all of the information and footnotes required by generally
accepted accounting principles for complete financial statements.  In the
Trustee's opinion, all adjustments necessary for a fair presentation have been
included.  For further information, refer to the financial statements and
footnotes included in the Trust's annual report on Form 10-K for the year ended
December 31, 1997.

Summary of Significant Accounting Policies

                 The accompanying unaudited financial statements have been
                 prepared on the following basis:

                 (a)      The royalty income recorded for a month is the amount
                          computed and paid by the lease holder to the Trustee
                          for such month.

                 (b)      Trust administration expenses are recorded in the
                          month they accrue.

This basis for reporting royalty income is thought to be the most meaningful
because distributions to the Unit Holders for a month are based on net cash
receipts for such month.  However, these statements differ from financial
statements prepared in accordance with generally accepted accounting principles
because, under such principles, royalty income for a month would be based on
production for such month without regard to when calculated or received.  In
addition, amortization of the net royalty and overriding royalty interests,
which is calculated on a units-of-production basis by lease, is charged
directly to trust corpus, since such amount does not affect distributable
income.





                                     - 4 -
<PAGE>   5


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

         a.      Material Changes in Financial Condition.

                 Kiewit Royalty Trust is a royalty trust with royalty and
overriding royalty interests in certain coal leases.  The Trust was formed for
the purposes of (1) administering the income received from such coal leases and
(2) distributing such income (together with interest earned thereon less
payment of or provision for obligations) to the holders of the Units of
Beneficial Interest.

         b.      Material Changes in Results of Operations.

                 During the first quarter of 1998, the Trust received a total
$2,534,970 of royalty and overriding royalty payments.  The following schedule
reflects the royalty and overriding royalty amounts received by the Trust from
each mine for the three months ended March 31, 1998 and 1997.

<TABLE>
<CAPTION>
                 Mine                                First Quarter
                 ----                                -------------
                                               1998                                  1997
                                               ----                                  ----
                 <S>                       <C>                                    <C>
                 Decker                    $ 2,470,670                            $1,837,048
                 Big Horn                       64,300                                13,057
                 Black Butte                     ---                                   ---
                 Spring Creek                    ---                                   ---
                                                                                                                       

                                           ------------                           ------------
                                           $2,534,970                             $1,850,105
                                           ==========                             ==========

</TABLE>

                 i.       Decker Mine.

                 The amount of royalties and overriding royalties received by
the Trust with respect to the Decker Mine increased to $2,470,670 in the first
quarter of 1998 compared to $1,837,048 during the same period in 1997.  The
increase for the first quarter of 1998 was due to changes in the relative
amounts of coal mined under leases bearing high and low overriding royalty
rates per ton, which are a normal result of the execution of a mining plan
encompassing several coal leases bearing different royalty rates and increased
production from the applicable leases.

                 ii.      Big Horn Mine.

                 Royalties received from the Big Horn Mine increased to $64,300
during the first quarter of 1998 from $13,057 in 1997.  This increase reflects
increased production from the applicable lease.





                                     - 5 -
<PAGE>   6
                 iii.     Black Butte Mine.

                 The Trust received no royalties from the Black Butte Mine
during the first quarter of 1998 and 1997.  This reflects the fact that Federal
Lease Number W-6266 was renewed effective April 1, 1996, and the Trust no
longer receives any overriding royalty on coal produced after March 31, 1996 as
explained below.

                 Lease number W-6266 provides that the terms and conditions of
the lease are subject to "reasonable readjustment" effective on April 1, 1996.
At that time, the royalty rate payable to the United States was increased to
12- 1/2%, which is the currently prevailing federal royalty rate.  Because of
the cap on the amount of the overriding royalty payable to the Trust with
respect to this lease, the increased royalty rate has effectively eliminated
further payments to the Trust with respect to this lease.

                 iv.      Spring Creek Mine.

                 No royalties were received from the Spring Creek Mine during
the first quarters of 1998 or 1997 because royalties are typically paid by the
mine operators on an annual basis during the second half of a calendar year.

                                    PART II
                               OTHER INFORMATION

ITEM 1.  LEGAL PROCEEDINGS.

         There are no material pending legal proceedings to which the Trust is
a party or of which any of its property is the subject.

ITEM 6.  EXHIBITS AND REPORTS ON FORM 8-K.

         (a)     Exhibits.

                 4.1      Kiewit Royalty Trust Indenture dated May 17, 1982, as
                          amended June 9, 1982 and June 23, 1982 (filed as
                          Exhibit 1 to the Trust's Form 10-Q filed with the
                          Securities and Exchange Commission on December 23,
                          1982, and incorporated herein by reference).

                 4.2      Order dated September 23, 1994, of the County Court
                          of Douglas County, Nebraska (filed as Exhibit 2 to
                          the Trust's Form 10-Q filed with the Securities and
                          Exchange Commission on November 14, 1994, and
                          incorporated herein by reference).





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<PAGE>   7

                 27.1     Financial Data Schedule.

        (b)      No reports on Form 8-K were filed during the quarter for which
                 this report is filed.





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<PAGE>   8
                                   SIGNATURE

                 Pursuant to the requirements of the Securities Exchange Act of
1934, the registrant has duly caused this report to be signed on its behalf by
the undersigned thereunto duly authorized.


Date: May 8, 1998                       KIEWIT ROYALTY TRUST
                                        --------------------
                                             (Registrant)

                                        By:  U.S. Bank National Association
                                             in its capacity as Trustee and 
                                             not in its individual capacity or
                                             otherwise 



                                             By: /s/  Susan K. Rosburg
                                                 ------------------------------
                                                      Susan K. Rosburg
                                                      Trust Officer





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