Robust underwater motion planning of autonomous underwater vehicles (AUVs) in dynamic cluttered environments is a problem that has yet to be addressed in depth. Due to advances in technology and computational capacity, AUVs are expected to operate safely and autonomously in increasingly challenging environments, necessitating methods that are able to safely navigate robots in real-time. Though, most solutions remain overly cautious and conservative. This paper proposes RUMP, a novel locally-optimal motion planning framework for robust real-time autonomous underwater navigation in 3D cluttered environments consisting of observed static and dynamic obstacles. The problem is modeled using path optimization and can be solved in real-time with a common nonlinear solver. The constructed objective function allows deciding the local goal during optimization to both maximize safety within a planning horizon and minimize the expected distance to the target position. Furthermore, path safety is considered for the entire transition between consecutive states, utilizing a novel approach for continuous spatiotemporal collision checks. The proposed formulation provides safe performance even in environments with obstacles that may move orders of magnitude faster than the AUV itself. Simulation experiments, in different challenging scenarios of obstacles moving up to 100 times faster than the robot, showcase robustness and efficient real-time performance of more than 15 Hz.