Maintaining the maximum stiffness of components with as little material as possible is an overarching objective in computational design and engineering. It is well-established that in stiffness-optimal designs, material is aligned with orthogonal principal stress directions. In the limit of material volume, this alignment forms micro-structures resembling quads or hexahedra. Achieving a globally consistent layout of such orthogonal micro-structures presents a significant challenge, particularly in three-dimensional settings. In this paper, we propose a novel geometric algorithm for compiling stress-aligned hexahedral lattice structures. Our method involves deforming an input mesh under load to align the resulting stress field along an orthogonal basis. The deformed object is filled with a hexahedral grid, and the deformation is reverted to recover the original shape. The resulting stress-aligned mesh is used as basis for a final hollowing procedure, generating a volume-reduced stiff infill composed of hexahedral micro-structures. We perform quantitative comparisons with structural optimization and hexahedral meshing approaches and demonstrate the superior mechanical performance of our designs with finite element simulation experiments.@en