The ultraviolet background (UVB) emitted by quasars and galaxies governs the ionization and thermal state of the intergalactic medium (IGM), regulates the formation of high-redshift galaxies, and is thus a key quantity for modeling cosmic reionization. The vast majority of cosmological hydrodynamical simulations implement the UVB via a set of spatially uniform photoionization and photoheating rates derived from UVB synthesis models. We show that simulations using canonical UVB rates reionize and, perhaps more importantly, spuriously heat the IGM, much earlier (z~15) than they should. This problem arises because at z>6, where observational constraints are nonexistent, the UVB amplitude is far too high. We introduce a new methodology to remedy this issue, and we generate self-consistent photoionization and photoheating rates to model any chosen reionization history. Following this approach, we run a suite of hydrodynamical simulations of different reionization scenarios and explore the impact of the timing of reionization and its concomitant heat injection on the thermal state of the IGM. We present a comprehensive study of the pressure smoothing scale of IGM gas, illustrating its dependence on the details of both hydrogen and helium reionization, and argue that it plays a fundamental role in interpreting Ly{alpha} forest statistics and the thermal evolution of the IGM. The premature IGM heating we have uncovered implies that previous work has likely dramatically overestimated the impact of photoionization feedback on galaxy formation, which sets the minimum halo mass able to form stars at high redshifts. We make our new UVB photoionization and photoheating rates publicly available for use in future simulations.