We report the results of a theoretical study of graphene/BN/graphene and BN/graphene/BN trilayers using the van-der-Waals-corrected density functional theory in conjunction with the non-equilibrium Green's Function method. These trilayer systems formed from graphene and BN exhibit distinct stacking-dependent features in their ground state electronic structure and response to an applied electric field perpendicular to the trilayer planes. The graphene/BN/graphene system shows a negligible gap in the electronic band structure that increases for the AAA and ABA stackings under an external electric field, while the zero-field band gap of BN/graphene/BN remains unaffected by the electric field. When both types of trilayer systems are contacted with gold electrodes, a metal-like conduction is predicted in the low-field regime, which changes to a p-type conduction with an increase in the applied perpendicular bias field.