We show how to exploit the rich hyperfine structure of fermionic alkali-metal atoms to produce a quasi-one-dimensional (quasi-1D) topological superfluid while avoiding excessive heating from off-resonant scattering. We model interacting fermions where four hyperfine states are coupled by a variety of optical and microwave fields. We calculate the local density of states in a trap, finding regimes with zero-energy topological edge modes. Heating rates in this system are significantly suppressed compared to simple Raman-induced spin-orbit coupling approaches. We also estimate the two- and three-body decay rates and find a reasonable lifetime at small but experimentally relevant densities. © 2014 American Physical Society.