Amenable groups that are left orderable

Let G be an abstract group. It is known that if every finitely generated subgroup of G has an infinite cyclic quotient, then G is left orderable. Using an idea of E.Ghys, we prove that the converse is true for all amenable groups. (This generalizes the known fact that the converse is true for all solvable groups.) The proof is surprisingly easy, and combines amenability with group theory, elementary topology, and the Poincare Recurrence Theorem (a basic result in the theory of dynamical systems).