The ability to organize nanoscale objects into well-defined three-dimensional (3D) arrays can translate advances in nanoscale synthesis into targeted material fabrication. Despite successes in nanoparticle assembly, most extant methods are system specific and not fully compatible with biomolecules. Here, we report a platform for creating distinct 3D ordered arrays from different nanomaterials using DNA-prescribed and valence-controlled material voxels. These material voxels consist of 3D DNA frames that integrate nano-objects within their scaffold, thus enabling the object's valence and coordination to be determined by the frame's vertices, which can bind to each other through hybridization. Such DNA material voxels define the lattice symmetry through the spatially prescribed valence decoupling the 3D assembly process from the nature of the nanocomponents, such as their intrinsic properties and shapes. We show this by assembling metallic and semiconductor nanoparticles and also protein superlattices. We support the technological potential of such an assembly approach by fabricating light-emitting 3D arrays with diffraction-limited spectral purity and 3D enzymatic arrays with increased activity.