We present an efficient method for computing thermal reaction rate constants that can be applied to systems in which transitions from reactant to product are infrequent. The method can be applied to high-dimensional, disordered systems which exhibit too many transition states to be identified, and for which useful reaction coordinates cannot be easily defined. The focus of our method is the time correlation function C(t), the normalized partition function for trajectories that begin in the reactant region and end in the product region after a time t; the time derivative of C(t) is the reaction rate constant, k(t). We use an umbrella potential to select initial positions from improbable regions of the reactant configuration space. We then compute C(t) directly by choosing random thermal momenta and asking if the resulting dynamical trajectory reaches the product region in time t. Since dynamical trajectories are run on the true potential energy surface, without the umbrella, re-crossing effects are included correctly. The initial condition bias introduced by the umbrella is removed by a weighting factor. We test the method on a simple two dimensional model potential and on a model for the isomerization of a diatomic in a Weeks-Chandler-Andersen fluid, and show that it gives accurate and precise rates with substantial reduction in computer time. (C) 2003 American Institute of Physics.