LiDAR Odometry is an essential component in many robotic applications. Unlike the mainstreamed approaches that focus on improving the accuracy by the additional inertial sensors, this letter explores the capability of LiDAR-only odometry through a continuous-time perspective. Firstly, the measurements of LiDAR are regarded as streaming points continuously captured at high frequency. Secondly, the LiDAR movement is parameterized by a simple yet effective continuous-time trajectory. Therefore, our proposed Traj-LO approach tries to recover the spatial-temporal consistent movement of LiDAR by tightly coupling the geometric information from LiDAR points and kinematic constraints from trajectory smoothness. This framework is generalized for different kinds of LiDAR as well as multi-LiDAR systems. Extensive experiments on the public datasets demonstrate the robustness and effectiveness of our proposed LiDAR-only approach, even in scenarios where the kinematic state exceeds the IMU's measuring range. Our implementation is open-sourced on GitHub.