We study bulk fermion fields in the localized gravity model with non-factorizable metric recently proposed by Randall and Sundrum, and Gogberashvili. In addition to a tower of weak-scale Kaluza-Klein states we find a zero mode for any value of the fundamental fermion mass. If the fermion mass is larger than half the curvature of the compact dimension, the zero mode can be localized on the ``hidden'' 3-brane in the Randall-Sundrum model. Identifying this mode with a right-handed neutrino provides a new way for obtaining small Dirac neutrino masses without invoking a see-saw mechanism. Cancellation of the parity anomaly requires introducing an even number of bulk fermions. This naturally leads to a strong hierarchy of neutrino masses and generically large mixing angles.