The rate of decay of correlations quantitatively describes the rate at which a chaotic system ``mixes'' the state space. We present a new rigorous method to estimate a lower bound for this rate of mixing. The technique may be implemented on a computer and is applicable to both multidimensional expanding and hyperbolic systems. The bounds produced are significantly less conservative than current rigorous bounds. In some situations it is possible to approximate the resonant eigenfunctions and to strengthen our bound to an estimate of the decay rate. Order of convergence results are stated.