The lattice cluster theory is extended to study all the possible nematic orderings of rectangular mesogens, namely, rodlike, discotic, and biaxial nematics for oblong mesogens, and rodlike and discotic nematics for square mesogens. The theory describes both the anisotropic hard core excluded volume and the anisotropic attractive van der Waals interactions on an equal footing. The partition function is expanded in terms of cluster contributions from the packing induced correlations and van der Waals interactions, with a Flory-type approximation as the zeroth-order treatment. The correlation contributions to the free energies for the isotropic liquid and the various homogeneous liquid crystalline phases are evaluated to second order in mesogen density and to first order in interaction energy. This theory therefore describes the anisotropic character of both the attractive and repulsive portions of the mesogen-mesogen interactions. Models with greater dimensionality provide greater orientational freedom and are shown to produce qualitatively different phase diagrams. As expected, the phase diagram is a strong function of the mesogen dimensions, and the placement of the biaxial phase with respect to the other phases is strongly dependent on mesogen size and the degrees of orientational freedom, in general accord with the nonuniversal behavior exhibited by experiments and computer simulations.