Colloidal self-assembly allows rational design of structures on the micrometer and submicrometer scale. One architecture that can generate complete three-dimensional photonic bandgaps is the diamond cubic lattice, which has remained difficult to realize at length scales comparable with the wavelength of visible or ultraviolet light. In this work, we demonstrate three-dimensional photonic crystals self-assembled from DNA origami that act as precisely programmable patchy colloids. Our DNA-based nanoscale tetrapods crystallize into a rod-connected diamond cubic lattice with a periodicity of 170 nanometers. This structure serves as a scaffold for atomic-layer deposition of high-refractive index materials such as titanium dioxide, yielding a tunable photonic bandgap in the near-ultraviolet.