This paper addresses the tracking control problem of 3D trajectories for underactuated underwater robotic vehicles operating in a constrained workspace including obstacles. More specifically, a robust Nonlinear Model Predictive Control (NMPC) scheme is presented for the case of underactuated Autonomous Underwater Vehicles (AUVs) (i.e., unicycle-like vehicles actuated only in surge, heave and yaw). The purpose of the controller is to steer the unicycle-like AUV to a desired trajectory with guaranteed input and state constraints (e.g., obstacles, predefined vehicle velocity bounds, thruster saturations) inside a partially known and dynamic environment where the knowledge of the operating workspace is constantly updated via the vehicle's on-board sensors. In particular, considering the sensing range of the vehicle, obstacle avoidance with any of the detected obstacles is guaranteed, by on-line generation of a collision-free trajectory tracking path, despite the model dynamic uncertainties and the presence of external disturbances representing ocean currents and waves. Finally, realistic simulation studies verify the performance and efficiency of the proposed framework.