We calculate the conductance through Aharonov-Bohm chain and ladder rings pierced by a magnetic flux which couples with the charge degrees of freedom. The system is weakly coupled to two leads and contains strongly interacting electrons modeled by the prototypical t - J and Hubbard models. For a wide range of parameters we observe characteristic dips in the conductance as a function of magnetic flux which are a signature of spin and charge separation. We also show how the dips evolve when the parameters of the models depart from the ideal case of total spin-charge separation. The ladder ring can be mapped onto an effective model for large anisotropy which can be easily analyzed. These results open the possibility of observing this peculiar many-body phenomenon in anisotropic ladder systems and in real nanoscopic devices.