  The physics responsible for gauge coupling unification may also induce small
neutrino masses. We propose a novel gauge mediated radiative seesaw mechanism
for calculable neutrino masses. These arise from quantum corrections mediated
by new $SU(3)_c \times SU(2)_L \times U(1)_X$ (3-3-1) gauge bosons and the
physics driving gauge coupling unification. Gauge couplings unify for a 3-3-1
scale in the TeV range, making the model directly testable at the LHC.
