  Casimir pistons are models in which finite Casimir forces can be calculated
without any suspect renormalizations. It has been suggested that such forces
are always attractive, but we present several counterexamples, notably a simple
type of quantum graph in which the sign of the force depends upon the number of
edges. We also show that Casimir forces in quantum graphs can be reliably
computed by summing over the classical orbits, and study the rate of
convergence of the periodic orbit expansion. In generic situations where no
analytic expression is available, the sign and approximate magnitude of Casimir
forces can often be obtained using only the shortest classical orbits.
