  It is shown how to obtain recently-proposed two-zero Majorana mass textures
in models with three Higgs triplets with small VEVs and a sufficiently massive
triplet Majoron by using abelian discrete symmetries. It is briefly discussed
how in SU(5) grand unification where the triplets occur in {\bf 15}'s the
neutrino textures can be related to up- and down- quark mass textures.
