This paper proposes a systematic anti-swing motion-planning method for three-dimensional overhead cranes, based on the load-swing dynamics of a two-dimensional overhead crane. First, a model-following anti-swing control law is designed based on the load-swing dynamics of a two-dimensional overhead crane, where the Lyapunov stability theorem is used as a mathematical tool. Then a new anti-swing motion-planning scheme is designed for a two-dimensional overhead crane based on the model-following anti-swing control law and typical crane operation in practice. Finally, the new anti-swing motion-planning scheme is extended for a three-dimensional overhead crane, based on the geometric relationship between a three-dimensional overhead crane and its two-dimensional counterpart. As a result, the proposed method avoids solving the load-swing dynamics of a three-dimensional overhead crane which is much more complicated than that of its two-dimensional counterpart. Furthermore, the proposed method can be applied to any existing overhead cranes without increasing their actuator torque capacity. The effectiveness of the proposed method is demonstrated by generating high-performance anti-swing trajectories with high-speed long-distance load hoisting.