Since the introduction of the Burridge-Knopoff model for fault mechanics many spring-block models have been proposed which purportedly generate a Gutenberg-Richter (GR) distribution of earthquake moments. Upon close examination, not all of these models generate distributions consistent with GR. Of those that do, few are able to generate other realistic aspects of distributed seismicity such as numbers of foreshocks and aftershocks consistent with Omori's Law and both characteristic and noncharacteristic seismic cycle behavior. We have systematically constructed a structurally-heterogeneous spring-block model coupled to a viscous asthenosphere which generates many realistic features of distributed seismicity. Versions of the model without viscous coupling were found to be inadequate to generate realistic model seismicity. A structurally heterogeneous model with viscous coupling is compared in detail to previous spring-block models and data for observed seismicity in terms of the cumulative frequency-size distribution of events, the occurrence of foreshocks and aftershocks, the increase of cumulative Benioff strain release prior to large earthquakes, the dependence of seismicity on depth, fault length, and structural heterogeneity, the relationship between the average slip and moment of an earthquake, and the heterogeneity of the rupture process. These comparisons demonstrate the effectiveness of the model in simulating seismicity and suggest that heterogeneity and viscous coupling are necessary conditions for a realistic spring-block model of seismicity.