The exact solution for a reaction-diffusion front propagating in a heterogeneous system of discrete, point-like sources is obtained without resorting to a representation of the sources by a spatially continuous function. When the reaction time is smaller than the characteristic diffusion time between neighboring sources, the front speed predicted by this discrete source model differs from the continuum theory based on the spatial averaging of the heterogeneities. Furthermore, when the sources are regularly distributed in space, discreteness introduces a limit and propagation beyond this limit is only possible in a system with randomly distributed sources via local fluctuations of the concentration. The discrete regime of front propagation is observed experimentally in suspensions of iron particles burning in oxygen-xenon mixtures.