  A unit vector field on a Riemannian manifold $M$ is called geodesic if all of
its integral curves are geodesics. We show, in the case of $M$ being a flat
3-manifold not equal to $\mathbb{E}^3$, that every such vector field is tangent
to a 2-dimensional totally geodesic foliation. Furthermore, it is shown that a
geodesic vector field $X$ on a closed flat 3-manifold is (up to rescaling) the
Reeb vector field of a contact form if and only if there is a contact structure
transverse to $X$ that is given as the orthogonal complement of some other
geodesic vector field. An explicit description of the lifted contact structures
(up to diffeomorphism) on the 3-torus is given in terms of the volume of $X$.
Finally, similar results for non-closed flat 3-manifolds are discussed.
