We investigate directed motion in nonadiabatically rocked ratchet systems sustaining few bands below the barrier. Upon restricting the dynamics to the lowest M bands, the total system-plus-bath Hamiltonian is mapped onto a discrete tight-binding model containing all the information both on the intrawell and interwell tunneling motion. A closed form for the current in the incoherent tunneling
regime is obtained. In effective single-band ratchets, no current rectification occurs. We apply our theory to describe rectification effects in vortex quantum ratchets devices. Current reversals upon variation of the ac-field amplitude or frequency are predicted.