We present a new method for calculating tunneling probabilities for chemical reactions with arbitrary curvature of the reaction path. The computational effort for obtaining a reaction probability at one energy consists of an integral over tunneling amplitudes for paths starting at various points on the reaction coordinate; for each point along the reaction coordinate, a one-dimensional search is performed to find the optimal tunneling path starting at that point; and for each tunneling path, a one-dimensional imaginary-action integral is evaluated. The method is designed to be applicable and practical even for general polyatomic reactions where no other reliable approach is affordable. To ascertain the accuracy of the method we have applied it to a wide range of one- and three-dimensional atom-diatom reactions on analytic potential energy surfaces for which accurate quantum mechanical rate constants are available. The accuracy, as compared to the accurate quantal calculations, is better than any previously available method that is simple enough to be applied to general polyatomic reactions.