The numerical solution of stochastic partial differential equations (SPDE) presents challenges not encountered in the simulation of PDEs or SDEs. Indeed, the roughness of the noise in conjunction with nonlinearities in the drift typically make these equations particularly stiff. In practice, this means that it is tricky to construct, operate, and validate numerical methods for SPDEs. This is especially true if one is interested in path-dependent expected values, long-time simulations, or in the simulation of SPDEs whose solutions have constraints on their domains. To address these numerical issues, this paper introduces a Markov jump process approximation for SPDEs, which we refer to as the spectral random walk method (SPECTRWM). The accuracy and ergodicity of SPECTRWM are verified in the context of a heat and overdamped Langevin SPDE, respectively. We also apply the method to Burgers and KPZ SPDEs.