Certain periodic dielectric structures can prohibit the propagation of light for all directions within a frequency range. These 'photonic crystals' allow researchers to modify the interaction between electromagnetic fields and dielectric media from radio to optical wavelengths. Their technological potential, such as the inhibition of spontaneous emission, enhancement of semiconductor lasers, and integration and miniaturization of optical components, makes the search for an easy-to-craft photonic crystal with a large bandgap, a major field of study. This progress article surveys a collection of robust complete three-dimensional dielectric photonic-bandgap structures for the visible and near-infrared regimes based on the diamond morphology together with their specific fabrication techniques. The basic origin of the complete photonic bandgap for the 'champion' diamond morphology is described in terms of dielectric modulations along principal directions. Progress in three-dimensional interference lithography for fabrication of near-champion diamond-based structures is also discussed.