Hysteresis loops and the associated avalanche statistics of spin systems, such as the random-field Ising and Edwards-Anderson spin-glass models, have been extensively studied. A particular focus has been on self-organized criticality, manifest in power-law distributions of avalanche sizes. Considerably less work has been done on the statistics of the times between avalanches. This paper considers this issue, generalizing the work of Nampoothiri et al. [Phys. Rev. E 96, 032107 (2017), 10.1103/PhysRevE.96.032107] in one space dimension to higher space dimensions. In addition to the interevent statistics of all avalanches, we also consider what happens when events are restricted to those exceeding a certain threshold size. Doing so raises the possibility of altering the definition of time to count the number of small events between the large ones, which provides for an analog to the concept of natural time introduced by the geophysics community with the goal of predicting patterns in seismic events. We analyze the distribution of time and natural time intervals both in the case of models that include only nearest-neighbor interactions, as well as models with (sparse) long-range couplings.