We imaged a 2^' × 2^' region of the Orion Nebula Cluster (ONC) in 1.3 mm wavelength continuum emission with the recently commissioned Combined Array for Research in Millimeter Astronomy (CARMA) and with the Submillimeter Array (SMA). Our mosaics include >250 known near-IR cluster members, of which 36 are so-called ``proplyds'' that have been imaged previously with the Hubble Space Telescope. We detected 40 sources in 1 mm continuum emission (one of which is the BN Object), and several of them are spatially resolved with our observations. The emission from most objects arises predominantly from dust, and circumstellar masses inferred for detected sources range from 0.01 to 0.5 M. The average circumstellar mass for undetected sources is estimated to be ~0.001 M, approximately an order of magnitude smaller than the minimum-mass solar nebula. Most stars in the ONC thus do not appear to currently possess sufficient mass in small dust grains to form Jupiter-mass (or larger) planets. Comparison with previous results for younger and older regions indicates that massive disks evolve significantly on ~Myr timescales. We also show that the percentage of stars in Orion surrounded by disks more massive than ~0.01 M is substantially lower than in Taurus, indicating that environment has an impact on the disk-mass distribution. Disks in Orion may be truncated through photoevaporation caused by the intense radiation field of the Trapezium stars, and we see marginal evidence for such a scenario in the spatial distribution of massive disks within the cluster. Our data show no statistically significant correlation between disk and stellar masses, although we see hints of a higher percentage of massive disks around lower mass stars.