Optoelectronic devices have long benefited from structuring in multiple dimensions on microscopic length scales. However, preserving crystal epitaxy, a general necessity for good optoelectronic properties, while imparting a complex three-dimensional structure remains a significant challenge. Three-dimensional (3D) photonic crystals are one class of materials where epitaxy of 3D structures would enable new functionalities. Many 3D photonic crystal devices have been proposed, including zero-threshold lasers, low-loss waveguides, high-efficiency light-emitting diodes (LEDs) and solar cells, but have generally not been realized because of material limitations. Exciting concepts in metamaterials, including negative refraction and cloaking, could be made practical using 3D structures that incorporate electrically pumped gain elements to balance the inherent optical loss of such devices. Here we demonstrate the 3D-template-directed epitaxy of group III-V materials, which enables formation of 3D structured optoelectronic devices. We illustrate the power of this technique by fabricating an electrically driven 3D photonic crystal LED.