  In this paper we find sufficient conditions for the vanishing of the
Morse-Novikov cohomology on Riemannian foliations. We work out a Bochner
technique for twisted cohomological complexes, obtaining corresponding
vanishing results. Also, we generalize for our setting vanishing results from
the case of closed Riemannian manifolds. Several examples are presented, along
with applications in the context of l.c.s. and l.c.K. foliations.
