Recent work on ultracold polar molecules, governed by a generalization of the

t-J Hamiltonian, suggests that molecules may be better suited than atoms for

studying d-wave superfluidity due to stronger interactions and larger

tunability of the system. We compute the phase diagram for polar molecules in a

checkerboard lattice consisting of weakly coupled square plaquettes. In the

simplest experimentally realizable case where there is only tunneling and an

XX-type spin-spin interaction, we identify the parameter regime where d-wave

superfluidity occurs. We also find that the inclusion of a density-density

interaction destroys the superfluid phase and that the inclusion of a

spin-density or an Ising-type spin-spin interaction can enhance the superfluid

phase. We also propose schemes for experimentally realizing the perturbative

calculations exhibiting enhanced d-wave superfluidity.