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We present results from extensive Monte Carlo (MC) simulations of domain growth in ferromagnets and binary mixtures with quenched disorder. These are modeled by the random-bond Ising model and the dilute Ising model with either nonconserved (Glauber) spin-flip kinetics or conserved (Kawasaki) spin-exchange kinetics. In all cases, our MC results are consistent with power-law growth with an exponent which depends on the quench temperature and the disorder amplitude . Such exponents arise naturally when the coarsening domains are trapped by energy barriers that grow logarithmically with the domain size. Our MC results show excellent agreement with the predicted dependence of .