We study the kinetics of ordering in Cu3Au using a model Hamiltonian derived from the effective medium theory of chemical bonding. Monte Carlo simulations are used to investigate universal and non-universal features of the growth kinetics. Anisotropic scaling of the structure factor is observed in late-stage growth of ordered domains. The anisotropy is a non-universal feature determined by details of the microscopic model, and we find that the anisotropy observed in the simulations is in excellent agreement with experiments on Cu3Au. The simulations are discussed in the context of theories of unstable growth. To our knowledge, this is the first study of kinetics in a realistic model Hamiltonian describing the material-specific properties of Cu3Au.