In this work, we reexamine sulfur chemistry occurring on and in the ice mantles of interstellar dust grains, and report the effects of two new modifications to standard astrochemical models: namely, (a) the incorporation of cosmic-ray-driven radiation chemistry and (b) the assumption of fast, nondiffusive reactions for key radicals in the bulk. Results from our models of dense molecular clouds show that these changes can have a profound influence on the abundances of sulfur-bearing species in ice mantles, including a reduction in the abundance of solid-phase H2S and HS, and a significant increase in the abundances of OCS, SO2, as well as pure allotropes of sulfur, especially S8. These pure-sulfur species-though nearly impossible to observe directly-have long been speculated to be potential sulfur reservoirs and our results represent possibly the most accurate estimates yet of their abundances in the dense interstellar medium. Moreover, the results of these updated models are found to be in good agreement with available observational data. Finally, we examine the implications of our findings with regard to the as-yet-unknown sulfur reservoir thought to exist in dense interstellar environments.