A new mimetic finite difference method for the diffusion problem is developed by using a linear interpolation for the numerical fluxes. This approach provides a higher-order accurate approximation to the flux of the exact solution. In analogy with the original formulation, a family of local scalar products is constructed to satisfy the fundamental properties of local consistency and spectral stability. The scalar solution field is approximated by a piecewise constant function. A computationally efficient postprocessing technique is also proposed to get a piecewise quadratic polynomial approximation to the exact scalar variable. Finally, optimal convergence rates and accuracy improvement with respect to the lower-order formulation are shown by numerical examples.