Recent works on 3D single object tracking treat the tracking as a target-specific 3D detection task, where an off-the-shelf 3D detector is commonly employed for tracking. However, it is non-trivial to perform accurate target-specific detection since the point cloud of objects in raw LiDAR scans is usually sparse and incomplete. In this paper, we address this issue by explicitly leveraging temporal motion cues and propose DMT, a Detector-free Motion prediction based 3D Tracking network that totally removes the usage of complicated 3D detectors, which is lighter, faster, and more accurate than previous trackers. Specifically, the motion prediction module is firstly introduced to estimate a potential target center of the current frame in a point-cloud free way. Then, an explicit voting module is proposed to directly regress the 3D box from the estimated target center. Extensive experiments on KITTI and NuScenes datasets demonstrate that our DMT, without applying any complicated 3D detectors, can still achieve better performance (~10% improvement on the NuScenes dataset) and faster tracking speed (i.e., 72 FPS) than state-of-the-art approaches. Our codes will be released publicly.