We develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non-planar faces. The material properties are described by a full tensor. We also prove superconvergence for the scalar (pressure) variable under very general assumptions. The theoretical results are confirmed with numerical experiments. In the practically important case of logically cubic meshes with randomly perturbed nodes, the mixed finite element with the lowest order Raviart-Thomas elements does not converge while the proposed mimetic method has the optimal convergence rate.