  Theories with large extra dimensions can generate small neutrino masses when
the standard model neutrinos are coupled to singlet fermions propagating in
higher dimensions. The couplings can also generate mass splittings and mixings
among the flavour neutrinos in the brane. We systematically study the minimal
scenario involving only one singlet bulk fermion coupling weakly to the flavour
neutrinos. We explore the neutrino mass structures in the brane that can
potentially account for the atmospheric, solar and LSND anomalies
simultaneously in a natural way. We demonstrate that in the absence of a priori
mixings among the SM neutrinos, it is not possible to reconcile all these
anomalies. The presence of some structure in the mass matrix of the SM
neutrinos can solve this problem. This is exemplified by the Zee model, which
when embedded in extra dimensions in a minimal way can account for all the
neutrino anomalies.
