  Alternatives to the see-saw mechanism are explored in supersymmetric models
with three right-handed or sterile neutrinos. Tree-level Yukawa couplings can
be drastically suppressed in a natural way to give sub-eV Dirac neutrino
masses. If, in addition, a B-L gauge symmetry broken at a large scale M_G is
introduced, a wider range of possibilities opens up. The value of the
right-handed neutrino mass M_R can be easily disentangled from that of M_G.
Dirac and Majorana neutrino masses at the eV scale can be generated radiatively
through the exchange of sneutrinos and neutralinos. Dirac masses m_D owe their
smallness to the pattern of light-heavy scales in the neutralino mass matrix.
The smallness of the Majorana masses m_L is linked to a similar see-saw pattern
in the sneutrino mass matrix. Two distinct scenarios emerge. In the first, with
very small or vanishing M_R, the physical neutrino eigenstates are, for each
generation, either two light Majorana states with mixing angle ranging from
very small to maximal, depending on the ratio m_D/M_R, or one light Dirac
state. In the second scenario, with a large value of M_R, the physical
eigenstates are two nearly unmixed Majorana states with masses \sim m_L and
\sim M_R. In both cases, the (B-L)-breaking scale M_G is, in general, much
smaller than that in the traditional see-saw mechanism.
