  We study the phase diagram of two models of spin-1/2 antiferromagnets
composed of corner-sharing tetrahedra, the basis of the pyrochlore structure.
Primarily, we focus on the Heisenberg antiferromaget on the checkerboard
lattice (also called the planar pyrochlore and crossed-chains model). This
model has an anisotropic limit, when the dimensionless ratio of two exchange
constants, J_\times/J << 1, in which it consists of one-dimensional spin chains
coupled weakly together in a frustrated fashion. Using recently developed
techniques combining renormalization group ideas and one-dimensional
bosonization and current algebra methods, we show that in this limit the model
enters a crossed dimer state with two-fold spontaneous symmetry breaking but no
magnetic order. We complement this result by an approximate ``quadrumer triplet
boson'' calculation, which qualitatively captures the physics of the
``plaquette valence bond solid'' state believed to obtain for J_\times/J = 1.
Using these known points in parameter space, the instabilities pointed to by
the quadrumer boson calculation, and the simple limit J_\times/J >> 1, we
construct a few candidate global phase diagrams for the model, and discuss the
nature of the quantum phase transitions contained therein. Finally, we apply
our quasi-one-dimensional techniques to an anisotropic limit of the
three-dimensional pyrochlore antiferromagnet, an approximate model for
magnetism in GeCu2O4. A crossed dimer state is predicted here as well.
