For circle foliations of compact $3$-manifolds, this is essentially answered by a theorem of Epstein: every such foliation is a Seifert fibration. Most Seifert fibrations are finitely covered by a product (surface)x(circle), and in these cases one of course has a free circle action.

One of the exceptional cases of Seifert fibrations not covered by a product is the fibration of $S^3$ obtained as follows. Take the linear foliation of slope p/q on a solid torus (completed by the core of the solid torus as another leaf), the linear foliation of slope q/p on another solid torus (again with the core as another leaf), and glue the two solid tori along their boundaries by sending one meridian to the longitude to the other and one longitude to the meridian of the other. The result is $S^3$ with a circle foliation, which does not seem to come from a free circle action if p and q are bigger than 1.