  We construct a family of rotationally invariant, local, S=1/2 Klein
Hamiltonians on various lattices that exhibit ground state manifolds spanned by
nearest-neighbor valence bond states. We show that with selected perturbations
such models can be driven into phases modeled by well understood quantum dimer
models on the corresponding lattices. Specifically, we show that the
perturbation procedure is arbitrarily well controlled by a new parameter which
is the extent of decoration of the reference lattice. This strategy leads to
Hamiltonians that exhibit i) $Z_2$ RVB phases in two dimensions, ii) U(1) RVB
phases with a gapless ``photon'' in three dimensions, and iii) a Cantor
deconfined region in two dimensions. We also construct two models on the
pyrochlore lattice, one model exhibiting a $Z_2$ RVB phase and the other a U(1)
RVB phase.
