This paper addresses an important materials engineering question: How can one identify the complete space (or as much of it as possible) of microstructures that are theoretically predicted to yield the desired combination of properties demanded by a selected application? We present a problem involving design of magnetoelastic Fe-Ga alloy microstructure for enhanced elastic, plastic and magnetostrictive properties. While theoretical models for computing properties given the microstructure are known for this alloy, inversion of these relationships to obtain microstructures that lead to desired properties is challenging, primarily due to the high dimensionality of microstructure space, multi-objective design requirement and non-uniqueness of solutions. These challenges render traditional search-based optimization methods incompetent in terms of both searching efficiency and result optimality. In this paper, a route to address these challenges using a machine learning methodology is proposed. A systematic framework consisting of random data generation, feature selection and classification algorithms is developed. Experiments with five design problems that involve identification of microstructures that satisfy both linear and nonlinear property constraints show that our framework outperforms traditional optimization methods with the average running time reduced by as much as 80% and with optimality that would not be achieved otherwise.