We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral grids. To approximate the flux (vector variable), the method uses two degrees of freedom per element edge in two dimensions and $n$ degrees of freedom per ($n$-gon) element face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen inner product in the space of discrete fluxes allows for local flux elimination and reduction of the method to a symmetric cell-centered finite difference scheme for the pressure. In the case of simplicial grids, optimal first-order convergence is proved for both variables, as well as second-order convergence for the scalar variable. Numerical results confirm the theory.