  We show that a closed orientable 3--dimensional manifold admits a round fold
map into the plane, i.e. a fold map whose critical value set consists of
disjoint simple closed curves isotopic to concentric circles, if and only if it
is a graph manifold, generalizing the characterization for simple stable maps
into the plane. Furthermore, we also give a characterization of closed
orientable graph manifolds that admit directed round fold maps into the plane,
i.e.\ round fold maps such that the number of regular fiber components of a
regular value increases toward the central region in the plane.
