We present a computational analysis of the influence of taking the full multiplet structure into account in calculations of the single-ion contribution to the magnetocrystalline anisotropy (MCA) of 3d transition-metal (TM) ions in localized-electron compounds at T=0 K. For atoms with dn (n=1-4,6-9) configurations, at sites with a local cubic, tetragonal, or trigonal symmetry the single ion MCA on the basis of the Hund's rule ground state term only [crystal field (CF) theory] is compared with the single-ion MCA on the basis of a fully relativistic first principles atomic theory including the intra-atomic d-d interaction. Solid state effects are taken into account by effective crystal fields and the exchange field. Under certain, realistic, conditions the use of the full multiplet theory is shown to have a significant effect on the calculated single-ion MCA. We also discuss the effect on the overall cubic anisotropy constants for cubic crystals containing transition metal atom sublattices for which the d-metal atoms are located on sites with a local tetragonal and trigonal symmetry. Possible refinements of the theory are discussed.