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A new three-dimensional (3D) periodic dielectric structure constructed with layers of dielectric rods of circular, elliptical, or rectangular shape is introduced. This new structure possesses a full photonic band gap of appreciable frequency width. At midgap, an attenuation of 21 dB per unit cell is obtained. This gap remains open for refractive indices n ≥ 1.9. Furthermore, this new 3D layer structure potentially has the additional advantage that it can be easily fabricated using conventional microfabrication techniques on the scale of optical wavelengths.