We study a mimetic finite difference discretization of diffusion-type problems on unstructured polyhedral meshes. We demonstrate the high accuracy of the approximate solutions for general diffusion tensor, the second order convergence rate for the scalar unknown and the first order convergence rate for the vector unknown on smooth or slightly distorted smooth meshes, on non-matching meshes and even on meshes with irregular-shaped polyhedra but flat faces. We show that in general, the meshes with non-flat faces require more than one flux unknown per mesh face to get optimal convergence rates.