Planning safe paths in 3D workspace for high DoF robotic systems, such as manipulators, is a challenging problem, especially when the environment is populated with the dynamic obstacles that need to be avoided. In this case the time dimension should be taken into account that further increases the complexity of planning. To mitigate this issue we suggest to combine safe-interval path planning (a prominent technique in heuristic search) with the randomized planning, specifically, with the bidirectional rapidly-exploring random trees (RRT-Connect) -- a fast and efficient algorithm for high-dimensional planning. Leveraging a dedicated technique of fast computation of the safe intervals, we end up with an efficient planner dubbed SI-RRT. We compare it with the state of the art and show that SI-RRT consistently outperforms the competitors both in runtime and solution cost.