Novel field effect transistors with suspended graphene gates are demonstrated. By incorporating mechanical motion of the gate electrode, it is possible to improve the switching characteristics compared to a static gate, as shown by a combination of experimental measurements and numerical simulations. The mechanical motion of the graphene gate is confirmed by using atomic force microscopy to directly measure the electrostatic deflection. The device geometry investigated here can also provide a sensitive measurement technique for detecting high-frequency motion of suspended membranes as required, e.g., for mass sensing.