We consider the low-temperature spin dynamics of substitutionally disordered ferromagnetic alloys. A nearest-neighbor Heisenberg Hamiltonian is assumed. The linear spin-wave approximation is made. However, the disorder is treated exactly by use of a computer-simulation technique which works for very large finite models. Densities of states, partial densities of states, and the scattering law are presented for spins diluted with nonmagnetic atoms and for mixtures of atoms having different exchange couplings. We give detailed results for the simple cubic lattice as well as representative cases of body-centered- and face-centered-cubic lattices. The over-all behavior is discussed in terms of the development of bands of excitations associated with each constituent as the concentrations are varied from the impurity limits. We also compare our calculations with analytical approximations based on effective-medium theory and on exact moments. © 1977 The American Physical Society.