High-resolution imaging technologies such as micro-computed tomography (CT) have not only revealed the intricate geometrical properties of biomedical materials and systems such as hard tissue engineering scaffolds, but also provided important data sources for the mechanical integrity of these systems, including implants. For the latter purpose, mechanical properties such as elasticity or strength need to be assigned to the voxels making up the CT scan. This is customarily done from back-analysis of mechanical tests performed on the entire scaffold structure, so as to downscale the overall mechanical response to that of an average solid voxel. Accordingly, local inhomogeneities remain unconsidered, and this motivates the review of a more fundamental, bottom-up-type strategy which unleashes, in an interdisciplinary fashion, important additional information “hidden” inside each and every voxel: Namely, the focus is set on resolving microstructural and nanostructural features located inside each and every voxel of a CT image, and on quantifying their effects on the voxel-specific material properties. For this purpose, the basics of X-ray physics, which allow for the translation of voxel-specific attenuation information into compositional quantities, are combined with micromechanical modeling. This provides a theoretically founded, rigorous link between material composition and mechanical properties. The feasibility and usefulness of this approach are highlighted by application examples concerning porosity and elasticity distributions throughout glass- and ceramic-based tissue engineering scaffolds, together with the results of quasi-static structural simulations performed on the geometrically and mechanically reconstructed objects. Thereafter, generalizations concerning hierarchical systems such as bone are discussed, together with the dynamic loading realm and the consideration of tissue anisotropy, also as far as interaction with implants is concerned.