The classical theory of transition radiation generated when charged particles travel through a dielectric-vacuum interface is reconsidered in order to include the effects of the diffraction of the particle beam occurring at the surface. If the surface is atomically smooth, coherent Bragg reflection and transmission occur as well as specular reflection and transmission. Each Bragg beam is shown to provide a source of classical radiation which will be called coherent surface bremsstrahlung and whose theoretical energy spectrum and angular distribution are calculated. Ordinary transition radiation is recovered and corresponds to the particular case of the transmitted-particle trajectory undisturbed by the surface. The coherent surface bremsstrahlung is shown to be particularly intense when the primary beam strikes the surface at glancing incidence. For metal surface, the maximum radiative yield of the specular beam occurs at the wavelength of the nonradiative surface plasmon. For a rough surface, diffuse particle scattering is also produced by the surface roughness which will correspondingly generate incoherent surface bremsstrahlung. Both coherent- and incoherent-surface-radiation mechanisms may be important for the interpretation of recently observed anomalous radiation emitted in glancing-incidence experiments.