We introduce and analyze a V-cycle multigrid algorithm for cell-centered finite difference methods applied to second-order elliptic boundary value problems. Unlike conventional cell-centered multigrid algorithms that use the natural injection operator for prolongation, we use a new prolongation operator whose energy norm we prove is bounded by 1 in the constant coefficient case and 1+Ch in the nonconstant case. We are thus able to use general finite element multigrid theory to conclude that the V-cycle either converges well or serves as a reasonably good preconditioner, respectively. While our theory does not establish optimal performance, our numerical experiments do show that the resulting algorithm converges much faster than the conventional schemes. In fact, these results show that the energy norm convergence factor is small and remains bounded uniformly in the finest mesh size, while that of the conventional algorithm grows.