Many species experience periodic mortality events, often in the form of winter, yet there is little understanding of how periodic mortality may affect population dynamics. We investigate the effects of periodic mortality on the population dynamics of a predator and its prey, both of which have multiple generations between mortality events. Using the simplest possible model of predator-prey interactions, we show that periodic mortality may produce complex dynamics including alternative stable cycles. Periodic mortality events may also create predator-prey cycles that rotate backwards, with prey densities lagged behind predator densities. The addition of random environmental variability to the model demonstrates that periodic mortality can be a strong driver of population variability, with the vestiges of deterministic cycles taking the form of eruptive patterns in the long-term dynamics of the predator-prey system. These results show that the long-term population dynamics of systems experiencing regular episodes of mortality may exhibit a wealth of patterns not seen in the absence of periodic mortality events.