A left-ordering on a group is a way of ordering its elements which is compatible with the group multiplication. Such orderings are of interest in topology in part due to connections with foliations and  Heegaard Floer homology - in particular, there is an outstanding conjecture relating taut foliations, L-spaces, and left-orderable fundamental groups of rational homology 3-spheres. We discuss the special case of manifolds obtained by Dehn surgery on a knot and investigate a technique for constructing left-orderings for such manifolds.