Megathrust geometric properties exhibit some of the strongest correlations with maximum earthquake magnitude in global surveys of large subduction zone earthquakes, but the mechanisms through which fault geometry influences subduction earthquake cycle dynamics remain unresolved. Here, we develop 39 models of sequences of earthquakes and aseismic slip (SEAS) on variably-dipping planar and variably-curved nonplanar megathrusts using the volumetric, high-order accurate code tandem to account for fault curvature. We vary the dip, downdip curvature and width of the seismogenic zone to examine how slab geometry mechanically influences megathrust seismic cycles, including the size, variability, and interevent timing of earthquakes. Dip and curvature control characteristic slip styles primarily through their influence on seismogenic zone width: wider seismogenic zones allow shallowly-dipping megathrusts to host larger earthquakes than steeply-dipping ones. Under elevated pore pressure and less strongly velocity-weakening friction, all modeled fault geometries host uniform periodic ruptures. In contrast, shallowly-dipping and sharply-curved megathrusts host multi-period supercycles of slow-to-fast, small-to-large slip events under higher effective stresses and more strongly velocity-weakening friction. We discuss how subduction zones' maximum earthquake magnitudes may be primarily controlled by the dip and dimensions of the seismogenic zone, while second-order effects from structurally-derived mechanical heterogeneity modulate the recurrence frequency and timing of these events. Our results suggest that enhanced co- and interseismic strength and stress variability along the megathrust, such as induced near areas of high or heterogeneous fault curvature, limits how frequently large ruptures occur and may explain curved faults' tendency to host more frequent, smaller earthquakes than flat faults.