We study finite-state, finite-action, discounted infinite-horizon Markov decision processes with uncertain correlated transition matrices in deterministic policy spaces. Existing robust dynamic programming methods cannot be extended to solving this class of general problems. In this paper, based on a robust optimality criterion, an approximate robust policy iteration using a multilayer perceptron neural network is proposed. It is proven that the proposed algorithm converges in finite iterations, and it converges to a stationary optimal or near-optimal policy in a probability sense. In addition, we point out that sometimes even a direct enumeration may not be applicable to addressing this class of problems. However, a direct enumeration based on our proposed maximum value approximation over the parameter space is a feasible approach. We provide further analysis to show that our proposed algorithm is more efficient than such an enumeration method for various scenarios.