  We investigate periodic straight-line orbits (SLO) in Hamiltonian force
fields using both direct and inverse methods. A general theorem is proven for
natural Hamiltonians quadratic in the momenta in arbitrary dimension and
specialized to two and three dimension. Next we specialize to homogeneous
potentials and their superpositions, including the familiar H\'enon-Heiles
problem. It is shown that SLO's can exist for arbitrary finite superpositions
of $N$-forms. The results are applied to a family of generalized H\'enon-Heiles
potentials having discrete rotational symmetry. SLO's are also found for
superpositions of these potentials.
