Radio observations of starburst regions in galaxies have revealed groups of compact nonthermal sources that we interpret as radiative supernova remnants expanding in the interclump medium of molecular clouds. Because of the high pressure in starburst regions, the interclump medium may have a density similar to 10(3) H atoms cm(-3) in a starburst nucleus like M82 and greater than or equal to 10(4) H atoms cm(-3) in an ultraluminous galaxy like Arp 220. In M82, our model can account for the sizes, the slow evolution, the high radio luminosities, and the low X-ray luminosities of the sources. We predict expansion velocities similar to 500 km s(-1), which is slower than the one case measured by VLBI techniques. Although we predict the remnants to be radiative, the expected radiation is difficult to detect because it is at infrared wavelengths and the starburst is itself very luminous; one detection possibility is broad [O I] 63 mum line emission at the positions of the radio remnants. The more luminous and compact remnants in Arp 220 can be accounted for by the higher molecular cloud density. In our model, the observed remnants lose most of the supernova energy to radiation. Other explosions in a lower density medium may directly heat a hot, low density interstellar component, leading to the superwinds that are associated with starburst regions.