We study spreading of reactions in random media and prove that homogenization takes place under suitable hypotheses.  That is, the medium becomes effectively homogeneous in the large-scale limit of the dynamics of solutions to the PDE.  Hypotheses that guarantee this include fairly general stationary ergodic KPP reactions, as well as homogeneous ignition reactions in up to three dimensions perturbed by radially symmetric impurities distributed according to a Poisson point process.  In contrast to the original (second-order) reaction-diffusion equations, the limiting "homogenized" PDE for this model are (first-order) Hamilton-Jacobi equations, and the limiting solutions are discontinuous functions that solve these in a weak sense.  A key ingredient is a novel relationship between spreading speeds and front speeds for these models (as well as a proof of existence of these speeds), which can be thought of as the inverse of a well-known formula in the case of periodic media, but we are able to establish it even for more general stationary ergodic media.