The rational design of anisotropic interaction patterns is a key step for programming colloid and nanoparticle self-assembly and emergent functions. Herein, we demonstrate a concept for harnessing the capabilities of 3D DNA origami for extensive supracolloidal self-assembly, and showcase its use for making truly monodisperse, patchy, divalent nanocuboids that can self-assemble into supracolloidal fibrils via programmable DNA hybridization. A change in the number of connector duplexes at the patches reveals that multivalency and cooperativity play crucial roles to enhance superstructure formation. We further show thermal and chemical switching of the superstructures as first steps towards reconfigurable self-assemblies. This concept lays the groundwork for merging monodisperse 3D DNA origami, featuring programmable patchiness and interactions, with nanoparticle self-assembly.