<PAGE>   1
                                    FORM 10-Q

                                  UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

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

                       For the quarter ended June 30, 1999

                           Commission File No. 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)

                                 (402) 348-6000
                 ----------------------------------------------
                (Registrant's telephone no. including area code)

       Indicate by check mark 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
                                              ---    ---

                    APPLICABLE ONLY TO CORPORATE REGISTRANTS

       Indicate the number of shares outstanding of each of the registrant's
classes of common stock, as of the latest practicable date.

       As of June 30, 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 June 30, 1999 and December 31, 1998

<TABLE>
<CAPTION>
ASSETS                                               1999                                   1998
------                                               ----                                   ----
<S>                                             <C>                                 <C>
Cash equivalents                                $ 246,669                           $    163,236

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

Less accumulated amortization                   $(105,518)                          $    (98,702)
                                                ---------                           -------------

Net royalty and overriding
royalty interests in coal leases                $  62,299                           $     69,115
                                                ---------                           -------------

Total Assets                                    $ 308,968                           $    232,351
                                                =========                           ============


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

Distributions payable to Unit Holders           $ 246,669                           $    163,236

Trust Corpus:  12,633,432
Units of Beneficial Interest
authorized and outstanding                      $  62,299                           $     69,115
                                                ---------                           ------------

Total Liabilities and Trust Corpus              $ 308,968                           $    232,351
                                                =========                           ============
</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 and six months ended June 30, 1999 and 1998

<TABLE>
<CAPTION>
                                                    Three Months Ended                             Six Months Ended
                                                         June 30                                      June 30
                                                    ------------------                            ------------------

                                                    1999                      1998               1999             1998
                                                    ----                      ----               ----             ----
<S>                                              <C>                  <C>               <C>                <C>
Royalty Income                                   $       253,698      $    203,430      $   2,028,179      $ 2,738,400

Interest Income                                            4,899             6,867             17,535           25,822

Trust Expenses                                           (11,928)          (12,374)           (30,389)         (30,756)
                                                    ------------      ------------      -------------      -----------

Distributable
   Income                                        $       246,669      $    197,923      $   2,015,325      $ 2,733,466

Distributable
   income per unit
   (12,633,432 units)                            $      .0195251      $   .0156666      $    .1595232      $  .2163677
</TABLE>

                      STATEMENT OF CHANGES IN TRUST CORPUS
                for the six months ended June 30, 1999 and 1998

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

Amortization of royalty interests                (6,816)                 (6,972)

Distributable Income                          2,015,325               2,733,466

Distribution to Unit Holders                ( 2,015,325)           (  2,733,466)
                                            ------------           ------------

Trust Corpus as of June 30                  $    62,299            $     75,234
                                            ===========            ============
</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 June 30, 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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<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 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 six months ended June 30, 1999, the Trust received a
total of $2,028,179 of royalty and overriding royalty payments; during the three
months ended June 30, it received a total of $253,698 of such payments. The
following schedule reflects the royalty and overriding royalty amounts received
by the Trust from each mine.

<TABLE>
<CAPTION>
                               Three Months Ended                           Six Months Ended
Mine                                June 30                                      June 30
----                   --------------------------------           --------------------------------------

                           1999                1998                   1999                   1998
                           ----                ----                   ----                   ----
<S>                      <C>                 <C>                   <C>                     <C>
Decker                    204,967             158,230               1,917,149              2,628,900

Big Horn                   48,731              45,200                 111,030                109,500

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

                         $253,698            $203,430              $2,028,179             $2,738,400
                         ========            ========              ==========             ==========
</TABLE>

        i.     DECKER MINE.

        The royalty and overriding royalty payments received by the Trust during
the first six months of 1999 with respect to the Decker Mine decreased to
$1,917,149 compared to $2,628,900 received during the same period in 1998. The
decrease in payments received in the first six months of 1999 was due to a
lower average selling price for the coal mined during the relevant period. For
the second quarter, the payments received by the Trust increased to $204,967
compared to $158,230 received during the same period in 1998. The increase in
payments received in the second quarter of 1999 was due to increased production
and a higher average selling price for the coal mined during the relevant
period.

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        ii.    BIG HORN MINE.

       Royalties received from the Big Horn Mine increased to $48,731 during the
second quarter of 1998 from $45,200 in 1998, and overall receipts for the first
six months of 1999 increased to $111,030 compared to $109,500 during the
corresponding period in 1998. These increases for 1999 reflect increased
production in the relative amounts of coal mined under leases bearing high and
low overriding royalty rates per ton.

        iii.   SPRING CREEK MINE.

       No royalties were received from the Spring Creek Mine during the first
six months 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 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




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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 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).

        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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                                    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.


                                   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

Dated: August 10, 1998



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