We present a theoretical study of photocurrents generated by direct optical transitions induced by linearly polarized radiation in monolayers of transition metal dichalcogenides. A microscopic theory of both the ballistic and shift contributions to the photocurrent is developed that accounts for Coulomb attraction between photoexcited electrons and holes. It is demonstrated that, in typical 2D semiconductors, the Coulomb correlation causes the ballistic contribution to dominate over the shift photocurrent.