In this paper, a new method for robot navigation in dynamic environments, called the reciprocal orientation algorithm, is introduced. This algorithm deals with the case in which each robot moves without direct communication with the other robots. The algorithm suggested in this paper uses the concept of reciprocal orientation which guarantees both smooth and collision-free robot trajectories. The algorithm is implemented in several simulation scenarios, some of them involving tens of robots. The deadlock problem, which occurs between two robots, has been solved by using an adequate deadlock resolution algorithm.