In this work we consider an agent trying to maximize a submodular reward function while moving in a graph environment. Such reward functions can be used to capture a variety of crucial sensing objectives in robotics including, but not limited to, mutual information and entropy. Furthermore, the agent must satisfy a mission specified by temporal logic constraints, which can encode many rich and complex missions such as 'visit regions A or B, then visit C, infinitely often. Never visit D before visiting C.' We present an algorithm to maximize a submodular reward function under these constraints and provide an approximation for the performance of the proposed algorithm. The results are validated via simulation.