We study a system of polar fermions in a two-dimensional optical lattice and show that the multiband Fermi-Hubbard model is necessary to discuss its properties. We take into account both onsite and long-range interactions between different bands, as well as occupation-dependent inter- and intraband tunnelings. For strong-enough dipolar interactions we predict the appearances of phases such as multiband crystals, smectic metal, and exotic p-wave supersolids.