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We investigate the general structure of orbital exchange physics inMott-insulating states of $p$-orbital systems in optical lattices. Orbitalorders occur in both the triangular and Kagome lattices. In contrast, orbitalexchange in the honeycomb lattice is frustrated as described by a novel quantum120$^\circ$-model. Its classical ground states are mapped into configurationsof the fully-packed loop model with an extra U(1) rotation degree of freedom.Quantum orbital fluctuations select a six-site plaquette ground state orderingpattern in the semiclassical limit from the ``order from disorder'' mechanism.This effect arises from the appearance of a zero energy flat-band of orbitalexcitations.

Orbital ordering and frustration of $p$-band Mott-insulators