We performed a comprehensive analysis of the extension of the discrete dipole approximation (DDA) to a rectangular cuboid lattice of dipoles. The theoretical analysis of two different approaches, based either on the point-dipole interaction or on the integration of Green's tensor (IGT), was performed starting with the rigorous integral equation for the electric field. We showed that the expressions for polarizability and interaction terms must strictly conform to each other, which resolves the existing controversy in the literature. Moreover, there are large differences between the spectra of the interaction matrix in the static limit for those DDA formulations. In particular, the point-dipole formulation leads to unphysical edges of the spectrum that deteriorate the convergence of the iterative solver with increasing refractive index. This severely limits the applicability of point-dipole DDA formulations with rectangular dipoles in contrast to the case of cubic dipoles. We implemented both above formulations in the open-source code ADDA and illustrated their performance on a number of test cases. In particular, we considered a graphene sheet, with thickness much smaller than the wavelength. The use of rectangular dipoles (with IGT) resulted in up to 100-times decrease of both simulation time and memory requirements, keeping the satisfactory accuracy. Similar improvements are expected for any strongly oblate or prolate particles in which the smallest dimension is much smaller than the wavelength.