Recent studies have demonstrated novel strategies for the organization of nanomaterials into three-dimensional (3D) ordered arrays with prescribed lattice symmetries using DNA-based self-assembly strategies. In one approach, the nanomaterial is sequestered into DNA origami frames or "material voxels" and then coordinated into ordered arrays based on the voxel geometry and the corresponding directional interactions based on its valency. While the lattice symmetry is defined by the valency of the bonds, a larger-scale morphological development is affected by assembly processes and differences in energies of anisotropic bonds. To facilely model this assembly process, we investigate the self-assembly behavior of hard particles with six interacting vertices via theory and Monte Carlo simulations and exploration of corresponding experimental systems. We demonstrate that assemblies with different 3D crystalline morphologies but the same lattice symmetry can be formed depending on the relative strength of vertex-to-vertex interactions in orthogonal directions. We observed three distinct assembly morphologies for such systems: cube-like, sheet-like, and cylinder-like. A simple analytical theory inspired by well-established ideas in the areas of protein crystallization, based on calculating the second virial coefficient of patchy hard spheres, captures the simulation results and thus represents a straightforward means of modeling this self-assembly process. To complement the theory and simulations, experimental studies were performed to investigate the assembly of octahedral DNA origami frames with varying binding energies at their vertices. X-ray scattering confirms the robustness of the formed nanoscale lattices for different binding energies, while both optical and electron microscopy imaging validated the theoretical predictions on the dependence of the distinct morphologies of assembled state on the interaction strengths in the three orthogonal directions.