In recent years, unmanned aerial vehicles (UAVs) are used for numerous inspection and video capture tasks. Manually controlling UAVs in the vicinity of obstacles is challenging, however, and poses a high risk of collisions. Even for autonomous flight, global navigation planning might be too slow to react to newly perceived obstacles. Disturbances such as wind might lead to deviations from the planned trajectories. In this work, we present a fast predictive obstacle avoidance method that does not depend on higher-level localization or mapping and maintains the dynamic flight capabilities of UAVs. It directly operates on LiDAR range images in real time and adjusts the current flight direction by computing angular potential fields within the range image. The velocity magnitude is subsequently determined based on a trajectory prediction and time-to-contact estimation. Our method is evaluated using Hardware-in-the-Loop simulations. It keeps the UAV at a safe distance to obstacles, while allowing higher flight velocities than previous reactive obstacle avoidance methods that directly operate on sensor data.