  We study and solve the ground-state problem of a microscopic model for a
family of orbitally degenerate quantum magnets. The orbital degrees of freedom
are assumed to have directional character and are represented by static
Potts-like variables. In the limit of vanishing Hund's coupling, the
ground-state manifold of such a model is spanned by the hard-core dimer (spin
singlet) coverings of the lattice. The extensive degeneracy of dimer coverings
is lifted at a finite Hund's coupling through an order-out-of-disorder
mechanism by virtual triplet excitations. The relevance of our results to
several experimentally studied systems is discussed.
