In cells, proteins rapidly self-assemble into sophisticated nanomachines. Bioinspired self-assembly approaches, such as DNA origami, have been used to achieve complex three-dimensional (3D) nanostructures and devices. However, current synthetic systems are limited by low yields in hierarchical assembly and challenges in rapid and efficient reconfiguration between diverse structures. Here, we developed a modular system of DNA origami "voxels" with programmable 3D connections. We demonstrate multifunctional pools of up to 12 unique voxels that can assemble into many shapes, prototyping 50 structures. Programmable switching of local connections between flexible and rigid states achieved rapid and reversible reconfiguration of global structures in three dimensions. Multistep assembly pathways were then explored to increase the yield. Voxels were assembled via flexible chain intermediates into rigid structures, increasing yield up to 100-fold. We envision that foldable chains of DNA origami voxels can achieve increased complexity in reconfigurable nanomaterials, providing modular components for the assembly of nanorobotic systems with future applications in synthetic biology, assembly of inorganic materials, and nanomedicine.