The Monte Carlo model is one of the most frequently used approaches to simulate grain growth, and retains a number of features that derive from the closely related Ising and Potts models. The suitability of these features for the simulation of grain growth is examined, and several modifications to the Hamiltonian and transition probability function are proposed. The resulting model is shown to not only reproduce the usual behaviors of grain growth simulations, but to substantially reduce the effect of the underlying pixel lattice on the microstructure as compared to contemporary simulations.