The inverse design of cellular materials that can meet multiple macroscale performance objectives constitutes a significant engineering challenge, primarily due to the high computational cost required for the exploration of broad design spaces. In the last decades, the development of machine learning methods along with the substantial enhancement of the computational limits has opened new frontiers in the analysis and design of materials. In the current work, machine learning methods are employed along with homogenization analysis techniques to inversely identify cellular material patterns that optimally meet the entire set of mechanical parameters contained within a fully populated elasticity tensor. The genetic algorithm modulates the inner material distribution to optimally meet both the stiffness objectives and relative density macroscale constraints posed. The approach allows for the identification of both isotropic and highly anisotropic cellular designs with positive, zero or auxetic Poisson’s ratio values. Its accuracy is verified using periodic, Abaqus-based finite element models. Moreover, dedicated 3D-printed specimens are manufactured and experimentally tested, highlighting its robustness and potential for wide-applicability.