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This paper presents a new method for efficient and exact collision-checking of linear motions of 3-D rigid bodies. 3-D rigid bodies have 6-D configuration spaces (three degrees of freedom for position and three for orientation), and constitute an important subclass of motion planning problems. Our method can be used with any collision-checker that is capable of performing linear transformations and distance computations on 3-D geometry. As previous work has shown, computing the distance between the rigid body in some configuration and the workspace obstacles immediately determines the collision-status of surrounding configurations. Using a recursive procedure one can then determine exactly whether an entire motion of the rigid body is collision-free. In this paper, we will show that by performing an optimally selected linear transformation on the workspace, the collision-status of rigid body motions can be determined using significantly fewer (costly) distance computations. Since collision-checking is often the computational bottleneck in sampling-based motion planning, our approach allows for significant performance improvements of algorithms such as PRM and RRT when planning for 3-D rigid bodies. We demonstrate the benefit of our approach when used in combination with RRT to construct a planning tree in an illustrative benchmark motion planning scenario.