The latitudinal diversity gradient (LDG), in which the number of species increases from the poles to the Equator, is one of the best-established patterns of life on Earth. The pattern of species-rich Tropics relative to species-poor temperate areas has been recognized for well over a century, but mechanisms for its genesis are still debated vigorously. We use simulations to assess the possibility that spatio-temporal climatic changes could have generated large-scale patterns of biodiversity as a function of only three biological processes—speciation, extinction, and dispersal—omitting adaptive niche evolution, diversity-dependence and coexistence limits. In our simulations, speciation occurred in response to range disjunction, and only when populations had been isolated for a sufficient period of time, whereas extinction occurred when a species could no longer access suitable sites. Our simulations generated clear LDGs that closely match empirical LDGs for three major vertebrate groups. Higher tropical diversity resulted primarily from higher rates of low-latitude speciation. This speciation was driven by spatio-temporal variation in precipitation at low latitudes, rather than in temperature. We therefore propose that spatio-temporal heterogenous precipitation change may have driven high rates of low-latitude speciation, contributing to LDGs. Overall, simulations show that major global biodiversity patterns can derive from the interaction of species’ niches (fixed a priori in our simulations) with dynamic climate across complex, existing landscapes, without the need to invoke biotic interactions or niche-related adaptations.