  We investigate nearest-neighbor valence-bond wave functions on bipartite
three-dimensional lattices. By performing large-scale Monte Carlo simulations,
we find that long-range magnetic order coexists with dipolar four-spin
correlations on the cubic lattice, this latter feature being reminiscent of the
Coulomb phase for classical dimers on the same lattice. Similar properties are
found for the lower-coordination diamond lattice. While this suggests that the
coexistence of magnetic order and dipolar four-spin correlations is generic for
bipartite three-dimensional lattices, we show that simple generalizations of
these wave functions can encode different ordering behaviors.
