Understanding how the Earth's biogeochemical cycles respond to environmental change is a prerequisite for the prediction and mitigation of the effects of anthropogenic perturbations. Microbial populations mediate key steps in these cycles, yet they are often crudely represented in biogeochemical models. Here, we show that microbial population dynamics can qualitatively affect the response of biogeochemical cycles to environmental change. Using simple and generic mathematical models, we find that nutrient limitations on microbial population growth can lead to regime shifts, in which the redox state of a biogeochemical cycle changes dramatically as the availability of a redox-controlling species, such as oxygen or acetate, crosses a threshold (a "tipping point"). These redox regime shifts occur in parameter ranges that are relevant to the present-day sulfur cycle in the natural environment and the present-day nitrogen cycle in eutrophic terrestrial environments. These shifts may also have relevance to iron cycling in the iron-containing Proterozoic and Archean oceans. We show that redox regime shifts also occur in models with physically realistic modifications, such as additional terms, chemical states, or microbial populations. Our work reveals a possible new mechanism by which regime shifts can occur in nutrient-cycling ecosystems and biogeochemical cycles, and highlights the importance of considering microbial population dynamics in models of biogeochemical cycles.