We show that in supersymmetric models with nontrivial flavor structure in the soft-breaking sector, one can have flavor changing neutral currents mediated by neutral Higgses. For&Dgr;S,&Dgr;B=2processes and for a generic form of the soft-breaking terms, we find that Higgs-mediated FCNC amplitudes increase very rapidly with tan &bgr; and can exceedSUSYbox contribution by up to two orders of magnitude whentan&hthinsp;&bgr;∼mt/mb.©1999 American Institute of Physics.