We develop two- and three-dimensional models for breakdown of metal-loaded dielectrics based on the breakdown of random arrays of perfectly conducting cylinders and spheres embedded in a uniform dielectric and placed in a uniform external electric field. We determine the breakdown field, breakdown-path geometry, and dielectric constant as a function of metal packing fraction. The computer solution of Laplace's equation in the random geometry uses truncated multipole expansions and the random packing configurations are generated by the Monte Carlo method. We compare the simulation results with exact lower bounds for the dielectric constant and scaling arguments for the breakdown field, which predict a linear relationship between the breakdown field and both the average surface-to-surface spacing between the metal particles and the minimum dielectric gap. Finally, we show that experimental results for inert rocket propellents are in excellent agreement with the scaling prediction.