We employ the concepts of band theory to describe the behavior of electromagnetic waves in three dimensionally periodic face-centered-cubic (fcc) dielectric structures. This can produce a ``photonic band gap'' in which optical modes, spontaneous emission, and zero-point fluctuations are all absent. In the course of a broad experimental survey, we have found that most fcc dielectric structures have ``semimetallic'' band structure. Nevertheless, we have identified one particular dielectric ``crystal'' which actually has a ``photonic band gap.'' This dielectric structure, consisting of 86% empty space, requires a refractive index contrast greater than 3 to 1, which happens to be readily obtainable in semiconductor materials.