We determine both the in-plane and out-of-plane dynamics of viscoelastic membranes separating two viscous fluids in order to understand microrheological studies of such membranes. We demonstrate the general viscoelastic signatures in the dynamics of shear; bending; and compression modes. We show that these modes remain independent in the presence of hydrodynamic interactions. The full response functions for motion both in-plane and out-of-plane are derived for the general case of viscoelastic films in contact with arbitrary viscous fluids. Specifically; we derive closed-form expressions for the in-plane longitudinal and transverse response functions for viscous membranes embedded in fluid media. We also find a screening of the otherwise two-dimensional character of the response to point forces due to the presence of the solvent.