Additively manufactured objects often exhibit directional dependencies in their structure due to the layered nature of the printing process. While this dependency has a significant impact on the object's functional performance, the problem of finding the best build orientation to maximize structural robustness remains largely unsolved. We introduce an optimization algorithm that addresses this issue by identifying the build orientation that maximizes the factor of safety (FS) of an input object under prescribed loading and boundary configurations. First, we conduct a minimal number of physical experiments to characterize the anisotropic material properties. Next, we use a surrogate-based optimization method to determine the build orientation that maximizes the minimum factor safety. The surrogate-based optimization starts with a small number of finite element (FE) solutions corresponding to different build orientations. The initial solutions are progressively improved with the addition of new solutions until the optimum orientation is computed. We demonstrate our method with physical experiments on various test models from different categories. We evaluate the advantages and limitations of our method by comparing the failure characteristics of parts printed in different orientations.