Supercrystals made of periodically arranged semiconductor quantum dots (QDs) are promising structures for nanophotonics applications due to almost unlimited degrees of freedom enabling fine tuning of their optical responses. Here we demonstrate broad engineering opportunities associated with the possibility of tailoring the energy bands of excitons in two-dimensional quantum-dot supercrystals through the alteration in the QD arrangement. These opportunities offer an unprecedented control over the optical properties of the supercrystals, which may be used as a versatile material base for advanced photonics devices on the nanoscale.