We propose a realization of the one-dimensional random dimer model and certain N-leg generalizations using cold atoms in an optical lattice. We show that these models exhibit multiple delocalization energies that depend strongly on the symmetry properties of the corresponding Hamiltonian, and we provide analytical and numerical results for the localization length as a function of energy. We demonstrate that the N-leg systems possess similarities with their one-dimensional ancestors but are demonstrably distinct. The existence of critical delocalization energies leads to dips in the momentum distribution that serve as a clear signal of the localization-delocalization transition. These momentum distributions are different for models with different group symmetries and are identical for those with the same symmetry.