The answer is negative. Take, for instance, the irrational foliation of the flat 2-torus by geodesics with the obvious ${\mathbb R}$ action via translations along leaves. 

Note that every fiber bundle is locally trivial (by definition), so this should not have been one of the assumptions, only nonexistence of periodic orbits. 

A necessary and sufficient condition for the ${\mathbb R}$ action to come from a fibration is that the action is proper, this follows, for instance, from the slice theorem.