I consider the class of depleted pyrochlore lattices of corner-sharing triangles, made by removing spins from a pyrochlore lattice such that every tetrahedron loses exactly one. Previously known examples are the hyperkagome and kagome staircase. I give criteria in terms of loops for whether a given depleted lattice can order analogous to the kagome state, and also show how the pseudo-dipolar correlations (due to local constraints) generalize to even the random depleted case.