[en] A gas of strongly interacting single-species (spinless) p-orbital fermionic atoms in 2D optical lattices is proposed and studied. Several interesting new features are found. In the Mott limit on a square lattice, the gas is found to be described effectively by an orbital exchange Hamiltonian equivalent to a pseudospin-1/2 XXZ model. For a triangular, honeycomb, or kagome lattice, the orbital exchange is geometrically frustrated and described by a new quantum 120 deg. model. We determine the orbital ordering on the kagome lattice, and show how orbital wave fluctuations select ground states via the order by disorder mechanism for the honeycomb lattice. We discuss experimental signatures of various orbital ordering