Fuglede-Putnam theorem is not true in general for EP operators on Hilbert spaces. We prove that under some conditions the theorem holds good. If the adjoint operation is replaced by Moore-Penrose inverse in the theorem, we get Fuglede-Putnam type theorem for EP operators -however proofs are totally different. Finally, interesting results on EP operators have been proved using several versions of Fuglede-Putnam type theorems for EP operators on Hilbert spaces.