In this paper, the enhanced terahertz radiation transformed from surface plasmon polaritons, excited by a uniformly moving electron bunch, in a structure consisting of a monolayer graphene supported on a dielectric grating with arbitrary profile is investigated. The results show that the grating profile has significant influence on the dispersion curves and radiation characteristics including radiation frequency and intensity. The dependence of dispersion and radiation characteristics on the grating shape for both the symmetric and asymmetric gratings is studied in detail. Moreover, we find that, for an asymmetric grating with certain profile, there exist two different diffraction types, and one of the two types can provide higher radiation intensity comparing to the other one. These results will definitely facilitate the practical application in developing a room-temperature, tunable, coherent and miniature terahertz radiation source.