Since the first photogeologic exploration of Mars, vast mounds of layered sediments found within the Valles Marineris troughs (interior layered deposits, ILDs) have remained unexplained. Recent spectroscopic results showing that these materials contain coarse-grained hematite and sulfate suggest that they are fundamentally similar to layered sulfate deposits seen elsewhere on Mars, and are therefore a key piece of the global aqueous history of Mars. In this work we constrain the origin of the ILDs by considering mass balance equations. One model involving formation of the ILDs by groundwater upwelling requires that a significant fraction of the global Martian sulfur budget was concentrated in the Valles Marineris at the time when the ILDs formed. It also necessitates high deposition and erosion rates in the Hesperian. We favor an alternative model in which the ILDs formed in a configuration similar to what is observed today through atmospherically driven deposition of ice, dust, and volcanogenic sulfuric acid. Such a model is easily compatible with the global sulfur budget, and does not require significant erosion rates or large volumes of liquid water. We propose that formation of sulfate-rich layered sediments on Mars was governed through time by volcanogenic SO2 and H2O emission rates and dust production against a backdrop of obliquity variation in a largely cold and dry climate.