The exotic Kitaev honeycomb model has recently been proposed to describe some of the magnetic interactions in the honeycomb iridates Na2IrO3 and Li2IrO3. We now show that analogues of the Kitaev exchange may be generated for iridates and related compounds on many other structures in two and three dimensions. These include the spinel-based pyrochlore iridate Ir2O4 as well as the hyperkagome iridate Na4Ir3O8, a spin liquid candidate. While forfeiting the exact honeycomb Kitaev spin liquid, the other Kitaev-Heisenberg models generalize the honeycomb alpha=1/2 exact solution to a unitary transformation structured by the Klein four-group, satisfiable on all lattices presented, yielding stripy-like magnetic order in two dimensions and alternating spin up/down planes in three dimensions. Luttinger-Tisza approximation phase diagrams find regions of strong quantum fluctuations around the exact phases. Schwinger fermion mean field theory suggests gapless chiral spin liquids at large Kitaev exchange. Schwinger boson mean field analyses on the two dimensional lattices correctly capture the fluctuation-free phase and finds a highly stable (kappa_c ~ 0.7) gapped spin liquid phase on the triangular lattice at small Kitaev coupling. We hope this work will help establish the Kitaev-type exchange within the lore of frustrated magnetism as a possible interaction yielding novel phases of heavy magnetic ions such as in the materials Ir2O4 and Na4Ir3O8.