  We present a simple approach to create a strong $p$-wave interaction for
fermions in an optical lattice. The crucial step is that the combination of a
lattice setup with different orbital states and $s$-wave interactions can give
rise to a strong induced $p$-wave pairing. We identify different topological
phases and demonstrate that the setup offers a natural way to explore the
transition from Kitaev's Majorana wires to two-dimensional $p$-wave
superfluids. We demonstrate how this design can induce Majorana modes at edge
dislocations in the optical lattice, and we provide an experimentally feasible
protocol for the observation of the non-Abelian statistics.
