The electronic structure of a realistic three-dimensional periodic random superlattice (SL) [(GaAs)m/(AlAs)n]l are calculated for the first time using the large-cluster recursion method within the tight-binding framework. It is found that the localized valance band-tail states of the random SL, which are almost independent of the AlAs concentration, extend to the band-gap region of the corresponding ordered SL. The central peaks in the Al s and Ga s states which vary with the Al concentration are studied, and comparison with the alloy AlxGa1-xAs is made. Our results provide a guide to the band-pp engineering of the random superlattices.