<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, 1999

                         Commission file number 0-10810

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

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

                                 Trust Division
                         U.S. Bank National Association
                             d/b/a First Bank, N.A.
                               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 10, 1999, there were 12,633,432 Units of Beneficial
Interest of the registrant outstanding.


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                                     PART I
                              FINANCIAL INFORMATION

ITEM 1. FINANCIAL STATEMENTS.

                              KIEWIT ROYALTY TRUST
                STATEMENT OF ASSETS, LIABILITIES AND TRUST CORPUS
                   as of March 31, 1999 and December 31, 1998

<TABLE>
<CAPTION>
ASSETS                                                   1999                            1998
------                                                   ----                            ----
<S>                                             <C>                             <C>
Cash equivalents                                $     1,768,656                 $       163,236

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

Less accumulated amortization                   $      (100,470)                $       (98,702)

Net royalty and overriding
royalty interests in coal leases                $        67,347                 $        69,115
                                                ---------------                 ---------------

Total Assets                                    $     1,836,003                 $       232,351
                                                ===============                 ===============


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

Distributions payable to Unit Holders           $     1,768,656                 $       163,236

Trust Corpus:  12,633,432
Units of Beneficial Interest
authorized and outstanding                      $        67,347                 $        69,115
                                                ---------------                 ---------------

Total Liabilities and Trust Corpus              $     1,836,003                 $       232,351
                                                ===============                 ===============
</TABLE>



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


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                              KIEWIT ROYALTY TRUST

                        STATEMENT OF DISTRIBUTABLE INCOME
               for the Three Months Ended March 31, 1999 and 1998

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

                                                         1999                           1998
                                                         ----                           ----
<S>                                             <C>                             <C>
Royalty Income                                  $     1,774,480                 $     2,534,970

Interest Income                                 $        12,637                 $        18,955

Trust Expenses                                  $       (18,461)                $       (18,382)
                                                ---------------                 ---------------

Distributable Income                            $     1,768,656                 $     2,535,543

Distributable Income per Unit
(12,633,432 Units Outstanding)                         0.139998                        0.200701
</TABLE>


                      STATEMENT OF CHANGES IN TRUST CORPUS
               for the Three Months Ended March 31, 1999 and 1998

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

                                                         1999                            1998
                                                         ----                            ----
<S>                                             <C>                             <C>
Trust Corpus as of January 1                    $        69,115                 $        82,206

Amortization of Royalty Interest                $        (1,768)                $        (4,227)

Distributable Income                            $     1,768,656                 $     2,535,543

Distribution to Unit Holders                    $    (1,768,656)                $    (2,535,543)
                                                ---------------                 ---------------

Trust Corpus as of March 31                     $        67,347                 $        77,979
                                                ===============                 ===============
</TABLE>



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

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                              KIEWIT ROYALTY TRUST

                          Notes to Financial Statements
                   as of March 31, 1999 and December 31, 1998

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, 1998. 

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.


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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 1999, the Trust received a total
$1,774,480 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, 1999 and 1998.

<TABLE>
<CAPTION>
          Mine                              First Quarter
          ----                              -------------
                                   1999                           1998
                                   ----                           ----
       <S>                      <C>                            <C>
       Decker                   $1,712,181                     $2,470,670

       Big Horn                     62,299                         64,300

       Spring Creek                  ----                           ----
                                ----------                     ----------

                                $1,774,480                     $2,534,970
                                ==========                     ==========
</TABLE>

              i.     Decker Mine.

              The amount of royalties and overriding royalties received by the
Trust with respect to the Decker Mine decreased to $1,712,181 in the first
quarter of 1999 compared to $2,470,670 during the same period in 1998. The
change was the net result due to (i) changes in the relative amounts of coal
mined under leases bearing high and low overriding royalty rates per ton, which
were a normal result of the execution of a mining plan encompassing several coal
leases bearing different royalty rates, and (ii) a lower average selling price
for coal mined in 1998 resulting from the expiration of a long-term coal
contract which was replaced by several other contracts.

              ii.    Big Horn Mine.

              Royalties received from the Big Horn Mine decreased to $62,299
during the first quarter of 1999 from $64,300 in 1998. This change reflects
decreased production from the applicable lease.


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<PAGE>   6



              iii.  Spring Creek Mine.

              No royalties were received from the Spring Creek Mine during the
first quarters of 1999 or 1998 because royalties are typically paid by the mine
operators on an annual basis during the second half of a calendar year.

       c.     Year 2000 Issue.

              The "Year 2000" computer problem has arisen because many computer
applications worldwide will not properly recognize the date change from December
31, 1999, to January 1, 2000, potentially causing production of erroneous data,
miscalculations, system failures and other operational problems.

              Since the Trust does not have any computer system of its own,
initiatives to address the Year 2000 issue has been undertaken by the Trustee,
U.S. Bank National Association, to evaluate the Year 2000 impact on its critical
computer hardware and software. The Trust does not incur any cost in the
Trustee's addressing of its Year 2000 problem. The Trustee has indicated to the
Trust that it has developed a strategic plan focusing on achieving Year 2000
compliance. Certain systems are being replaced, modified or reprogrammed to be
Year 2000 compliant. If necessary, new hardware and software is being purchased
to be Year 2000 compliant. Systems are being tested in an isolated environment
dedicated to Year 2000 testing. The Year 2000 project includes contingency plans
to mitigate potential delays or other problems, such as back-up solutions for
all mission-critical applications and business continuation plans for
significant vendors and other business partners. The Year 2000 efforts of the
Trustee and any related third parties are not within the Trust's control,
however, and their failure to remediate Year


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<PAGE>   7


2000 issues successfully could result in business disruption, increased
operating costs and increased credit risk for the Trust. At the present time, it
is not possible to determine whether any such events are likely to occur, or to
quantify any potential negative impact they may have on the Trust's future
results of operations and financial condition.

              Ultimately, the potential impact of the Year 2000 issue will
depend not only on the success of the corrective measures undertaken by the
Trustee, but also on the way in which the Year 2000 issue is addressed by
vendors, service providers, counterparties, utilities, governmental agencies and
other entities with which the Trustee and the Trust do business. These parties
are receiving communication from the Trustee to heighten their awareness of the
Year 2000 issue, to learn how the Trustee is addressing it and to evaluate any
likely impact on the Trustee and the Trust.

              The foregoing discussion regarding Year 2000 contains
forward-looking statements, which are based on management's best estimates
derived using various assumptions. These forward-looking statements involve
inherent risks and uncertainties, and actual results could differ materially
from those contemplated by such statements. Such material differences could
result in, among other things, business disruption, operational problems,
financial loss, legal liability and similar risks.

                                     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).


                                      -7-
<PAGE>   8


              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).

              27.1   Financial Data Schedule.

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


                                      -8-
<PAGE>   9


                                    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.

<TABLE>
<S>                                       <C>
Date: May 10, 1999                       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
</TABLE>



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