We study the effect of temperature cycling on the rate of Ostwald ripening (or coarsening) of spherical particles dispersed in a binary solution. A widespread view, which states a temperature cycle generally enhances the rate of Ostwald ripening by first dissolving the smallest particles (heating) and then regrowing the dissolved amount of material on the remaining particles (cooling), is shown to be inadequate as it does not include transient effects. On the basis of a simulation method that assumes mass transfer as the limiting growth mechanism, we show that each temperature cycle is followed by a significant relaxation of the particle-size distribution, during which the number of particles remains constant, and the average particle size decreases. The relaxation is shown to be crucial to obtain a linear scaling of the average particle radius cubed with the number of cycles applied (or time), which is the behavior generally observed for the evolution of ice crystals in cycling experiments on frozen aqueous solutions or frozen foods. We show the experimentally observed increase in the proportionality constant (or "coarsening rate") as compared to isothermal ripening, or the increase of the coarsening rate with increasing cycle frequency, can be reproduced convincingly only if some (transient) ripening is allowed to take place at the elevated temperature of each cycle. Our results thus suggest the effect of temperature cycling on Ostwald ripening is governed by a dissolution-ripening-regrowth-relaxation mechanism.