  We present a method of computing Casimir forces for arbitrary geometries,
with any desired accuracy, that can directly exploit the efficiency of standard
numerical-electromagnetism techniques. Using the simplest possible
finite-difference implementation of this approach, we obtain both agreement
with past results for cylinder-plate geometries, and also present results for
new geometries. In particular, we examine a piston-like problem involving two
dielectric and metallic squares sliding between two metallic walls, in two and
three dimensions, respectively, and demonstrate non-additive and non-monotonic
changes in the force due to these lateral walls.
