The neutrino mass matrix generated by the Zee (radiative) mechanism has zero (in general, small) diagonal elements and a natural hierarchy of the nondiagonal elements. The propagation in a medium of the neutrinos with the Zee mass matrix is determined by the existence of two resonances: one at the density determined by the highest mass scale, the other at zero density. The resonances are characterized by strongly different values of widths and level splittings which allow us to reduce the three-neutrino task to two-neutrino tasks. In the physically interesting cases the probabilities of flavor transitions as functions of neutrino energy can be represented as two-neutrino probabilities modulated by the effect of vacuum oscillations related to the small mass splitting. Possible applications of the results to the solar, supernova, atmospheric and relic neutrinos are discussed. A set of predictions is found which could identify the Zee mass matrix and therefore the corresponding mechanism of mass generation.