A major challenge in DNA nanotechnology is to extend the length scale of DNA structures from the nanoscale to the microscale to enable applications in cargo delivery, sensing, optical devices, and soft robotics. Self-assembly of DNA origami building blocks provides a promising approach for fabricating such higher-order structures. Inspired by self-assembly of patchy colloidal particles, researchers have recently begun to introduce patches of mutually attractive chemical moieties at designated sites on DNA origami to assemble them into complex higher-order architectures. However, designing such functionalized DNA origamis to target specific assembly structures is highly challenging because the underlying relationship between the building block design and assembly structure is very complex. Machine learning is especially well suited for such inverse-design tasks. In this work, we develop a coarse-grained model of DNA origami nanocubes grafted with hydrophobic brushes and employ the neural adjoint (NA) method to explore highly ordered target assemblies of such origamis, including checkerboard, honeycomb, and Kagome lattices. We envision that our design approach can be generalized to more complex designs and used to tailor structural properties to expand the application space of DNA nanotechnology.