Rate dependent, micro-structured mechanical metamaterials have been engineered to have a wide band of attenuated transmission over a desired frequency range. Enhancing the performance of these structures (by widening the attenuation band or decreasing transmitted energy within that band) can be achieved by layering metamaterials with small design variations to form functionally graded media. In this work, a base unit cell geometry consists of a 3D printed square frame with an H-shaped resonator inclusion is used. Optimizing the attenuation band for a layered system with this geometry involves coordinating many degrees of geometric freedom and material options across several layers. The possible number of combinations exceeds the feasible limit for manual search and an exact analytical solution does not exist. Genetic Algorithm (GA) and Bayesian Optimization (BO) approaches are applied to the design scenario to reduce the total number of combinations that need to be evaluated to locate the best performing metamaterial. A reduced order model (ROM) is assembled to decrease computation time and avoid finite element software during each function call. The approximated frequency response curve is then graded using a weighted performance metric to determine how well the material attenuates energy within a desired frequency band. Solutions from GA and BO methods are used to confirm ideal solutions and locate alternative solutions that were overlooked by one approach. Designing efficient functionally graded metamaterials enables practical application by reducing the number of layers needed to achieve attenuation, thereby decreasing the thickness and weight of the required material. Applying machine learning techniques to increasingly complex material design challenges advances engineered mechanical metamaterials towards applications for lenses, pressure/blast attenuation, passive vibration filtering, and sound proofing.