An alternative to the conventional seesaw mechanism is proposed to explain the origin of small neutrino masses in supersymmetric theories. The masses and couplings of the right-handed neutrino field are suppressed by supersymmetry breaking, in a way similar to the suppression of the Higgs doublet mass μ. New mechanisms for light Majorana and Dirac neutrinos arise, depending on the degree of suppression. Superpartner phenomenology is greatly altered by the presence of weak scale right-handed sneutrinos, which may have a coupling to a Higgs boson and a left-handed sneutrino. The sneutrino spectrum and couplings are quite unlike the conventional case - the lightest sneutrino can be the dark matter and predictions are given for event rates at upcoming halo dark matter direct detection experiments. Higgs boson decays and search strategies are changed. Copious Higgs boson production at hadron colliders can result from cascade decays of squarks and gluinos. ©2001 The American Physical Society.