  We discuss the ground state properties of a spin 1/2 magnetic ion with
threefold $t_{2g}$ orbital degeneracy on a highly frustrated pyrochlore
lattice, like Ti$^{3+}$ ion in B-spinel MgTi$_2$O$_4$. We formulate an
effective spin-orbital Hamiltonian and study its low energy sector by
constructing several exact-eigenstates in the limit of vanishing Hund's
coupling. We find that orbital degrees of freedom modulate the spin-exchange
energies, release the infinite spin-degeneracy of pyrochlore structure, and
drive the system to a non-magnetic spin-singlet manifold. The latter is a
collection of spin-singlet dimers and is, however, highly degenerate with
respect of dimer orientations. This ``orientational'' degeneracy is then lifted
by a magneto-elastic interaction that optimizes the previous energy gain by
distorting the bonds in suitable directions and leading to a tetragonal phase.
In this way a valence bond crystal state is formed, through the condensation of
dimers along helical chains running around the tetragonal c-axis, as actually
observed in MgTi$_2$O$_4$. The orbitally ordered pattern in the dimerized phase
is predicted to be of ferro-type along the helices and of antiferro-type
between them. Finally, through analytical considerations as well as numerical
ab-initio simulations, we predict a possible experimental tool for the
observation of such an orbital ordering, through resonant x-ray scattering.
