The band structures of several semiconductor alloys have been calculated by a method which takes into account the reduced local symmetry in the alloys. The alloy band structure is found from the weighted average of the band structures calculated for crystals formed from the different nearest-neighbor configurations that can occur in the alloys. In this way one avoids making the usual single-site approximation in which the effects of local atomic rearrangements on the alloy band structure are neglected. The results are compared with experiment in two ways. First the variation with composition of the splitting of the valence bands at \ensuremath{\Gamma} is found to be in excellent agreement with experiment for the three alloys ${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$As, ${\mathrm{Ga}}_{\mathrm{x}}$${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$As, and ${\mathrm{Ga}}_{\mathrm{x}}$${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$P which were studied. In addition, the bowing of the \ensuremath{\Gamma}, L, and X band edges and the alloy composition at which the direct-to-indirect band-gap crossovers occur are in good agreement with experiment.