The ability to structure matter at near-atomic scales has resulted in significant technological advances in computing, medicine, materials science and other areas. Current techniques to do this have important limitations, and alternatives might greatly expand the range of devices and materials we can make. Self-assembly of DNA-coated microparticles, or colloids, is one very promising alternative, where DNA acts as an exquisitely programmable ``smart glue'' between the colloids. Using computer simulations, we will explore how to coax colloids coated with multiple, varied DNA sequences into spontaneously forming precise, intricate assemblies. Our immediate focus will be to build microstructures that will facilitate the scalable fabrication of 3D photonic crystals, a promising material with which to build future all-optical computing devices. The innumerable potential pathways to direct such an assembly can be explored much more efficiently through computer simulation than is possible experimentally. Nevertheless, by carefully limiting simulations to the experimental constraints of today and the near-future, we hope to realize the assemblies we see computationally in experiments.