We have investigated the electromagnetic properties of a 3D wire mesh in a geometry resembling covalently bonded diamond. The frequency and wave vector dispersion show forbidden bands at those frequencies ${\ensuremath{\nu}}_{0}$, corresponding to the lattice spacing, just as dielectric photonic crystals do. But they have a new forbidden band which commences at zero frequency and extends, in our geometry, to $\ensuremath{\sim}\frac{1}{2}{\ensuremath{\nu}}_{0}$, acting as a type of plasma cutoff frequency. Wire mesh photonic crystals appear to support a longitudinal plane wave, as well as two transverse plane waves. We identify an important new regime for microwave photonic crystals, an effective medium limit, in which electromagnetic waves penetrate deeply into the wire mesh through the aid of an impurity band.