  Using a multidimensional cut-off technique, we obtain expressions for the
cut-off dependent part of the vacuum energy for parallelepiped geometries in
any spatial dimension d. The cut-off part yields nonrenormalizable hypersurface
divergences and we show explicitly that they cancel in the Casimir piston
scenario in all dimensions. We obtain two different expressions for the
d-dimensional Casimir force on the piston where one expression is more
convenient to use when the plate separation a is large and the other when a is
small (a useful $a \to 1/a$ duality). The Casimir force on the piston is found
to be attractive (negative) for any dimension d. We apply the d-dimensional
formulas (both expressions) to the two and three-dimensional Casimir piston
with Neumann boundary conditions. The 3D Neumann results are in numerical
agreement with those recently derived in arXiv:0705.0139 using an optical path
technique providing an independent confirmation of our multidimensional
approach. We limit our study to massless scalar fields.
