Spatially synchronized fluctuations in system state are common in physical and biological systems ranging from individual atoms to species as diverse as viruses, insects and mammals. Although the causal factors are well known for many synchronized phenomena, several processes concurrently have an impact on spatial synchrony of species, making their separate effects and interactions difficult to quantify. Here we develop a general stochastic model of predator-prey spatial dynamics to predict the outcome of a laboratory microcosm experiment testing for interactions among all known synchronizing factors: (1) dispersal of individuals between populations; (2) spatially synchronous fluctuations in exogenous environmental factors (the Moran effect); and (3) interactions with other species (for example, predators) that are themselves spatially synchronized. The Moran effect synchronized populations of the ciliate protist Tetrahymena pyriformis; however, dispersal only synchronized prey populations in the presence of the predator Euplotes patella. Both model and data indicate that synchrony depends on cyclic dynamics generated by the predator. Dispersal, but not the Moran effect, 'phase-locks' cycles, which otherwise become 'decoherent' and drift out of phase. In the absence of cycles, phase-locking is not possible and the synchronizing effect of dispersal is negligible. Interspecific interactions determine population synchrony, not by providing an additional source of synchronized fluctuations, but by altering population dynamics and thereby enhancing the action of dispersal. Our results are robust to wide variation in model parameters representative of many natural predator-prey or host-pathogen systems. This explains why cyclic systems provide many of the most dramatic examples of spatial synchrony in nature.