We analyze electromagnetic radiation from a bunch passing through a corrugated conductive planar structure. The wavelengths of the radiation are assumed to be much greater than the corrugation period. Under this approximation, the corrugated structure can be replaced with a smooth surface on which the so-called equivalent boundary conditions (EBC) are fulfilled. Here, we also assume that the structure is deeply corrugated, i.e. the structure depth is of the same order as an inverse wavenumber. Using the EBC we obtain a general solution and investigate it asymptotically. It is shown that two types of radiation are generated: volume radiation and surface waves. Both types of the radiations are analyzed in detail. In particular, we demonstrate that the radiation is highly sensitive to the structure depth. Phenomena under investigation can be useful for generating the electromagnetic radiation and for particle bunch diagnostics.