We propose a Monte Carlo method for efficiently sampling trajectories with fixed initial and final conditions in a system with discrete degrees of freedom. The method can be applied to any stochastic process with local interactions, including systems that are out of equilibrium. We combine the proposed path sampling algorithm with thermodynamic integration to calculate transition rates. We demonstrate our method on the well-studied two-dimensional Ising model with periodic boundary conditions, and show agreement with other results for both large and small system sizes. The method scales well with the system size, allowing one to simulate systems with many degrees of freedom, and providing complementary information with respect to other algorithms.