The radiative signatures of the nanoflare model for coronal heating are investigated. If an observed coronal loop is assumed to consist of many small strands that cannot be distinguished spatially by EUV or X-ray observations, we are able to calculate differential emission-measure profiles and filling factors for a range of heating models. In this picture the strands undergo continual heating and cooling, leading to a corona comprising strands with a broad range of temperatures and densities. Thus, observations over a range of temperatures will show a multithermal coronal structure. The cyclical heating-cooling leads inevitably to loops that are underdense and overdense at high and low temperatures, respectively, compared to what would be expected from static equilibrium models, and in addition, we show that differential emission-measure profiles with shallow slopes can be obtained, as reported in recent observations. The differences between filling factors that can be seen by broadband and narrowband instruments are explored. Loops with broadband filling factors near unity can still have small narrowband factors, and the narrowband factor is shown to be a strong function of the local temperature. Nanoflare energy distributions that are constant, flat, or power laws are considered. Power laws lead to wide distributions of temperatures and densities in the corona, and steep power laws lead to larger filling factors.