The problem of robust multi-hypothesis testing in the Bayesian setting is studied in this paper. Under the $m \geq 2$ hypotheses, the data-generating distributions are assumed to belong to uncertainty sets constructed through some moment functions, i.e., the sets contain distributions whose moments are centered around empirical moments obtained from some training data sequences. The goal is to design a test that performs well under all distributions in the uncertainty sets, i.e., a test that minimizes the worst-case probability of error over the uncertainty sets. Insights on the need for optimization-based approaches to solve the robust testing problem with moment constrained uncertainty sets are provided. The optimal (robust) test based on the optimization approach is derived for the case where the observations belong to a finite-alphabet. When the size of the alphabet is infinite, the optimization problem is infinite-dimensional and intractable, and therefore a tractable finite-dimensional approximation is proposed, whose optimal value converges to the optimal value of the original problem as the size of the dimension {of the approximation} goes to infinity. A robust test is constructed from the solution to the approximate problem, and guarantees on its worst-case error probability over the uncertainty sets are provided. Numerical results are provided to demonstrate the performance of the proposed robust test.