We present the foundations of a new approach to the Casimir effect based on classical ray optics. We show that a very useful approximation to the Casimir force between arbitrarily shaped smooth conductors can be obtained from knowledge of the paths of light rays that originate at points between these bodies and close on themselves. Although an approximation, the optical method is exact for flat bodies, and is surprisingly accurate and versatile. In this paper we present a self-contained derivation of our approximation, discuss its range of validity and possible improvements, and work out three examples in detail. The results are in excellent agreement with recent precise numerical analysis for the experimentally interesting configuration of a sphere opposite an infinite plane. ; Comment: Minor corrections. Submitted to Nucl. Phys. B