We have theoretically investigated the electron and hole energy spectra and light absorption in the three-dimensionally ordered quantum dots superlattices (QDS) made of the direct band-gap semiconductors. The calculations were performed for QDS of the rhombic symmetry with a substantial electron (hole) wave-function overlap using a one-band Hamiltonian for the electrons and six-band Hamiltonian for the holes. The obtained results were compared with the predictions of the simplified models for the uncoupled heavy, light, and split-off holes. It has been shown that the energy spectra of the electrons and holes in the ordered QDS are distinctively different from those in the single quantum dots (QD) or conventional quantum-well superlattices. The charge-carrier dispersion and localization are very sensitive to the quasicrystallographic directions defined by the dots, which play a role of the atoms in such QD supracrystal. We found that in the ordered QDS the oscillator strength for the interband optical transitions can be high for a relatively wide range of the photon energies. The obtained results are important for the proposed applications of QDS in solar cells, photodetectors, and thermoelectrics.