Conventional LiDAR-inertial odometry (LIO) or SLAM methods heavily rely on geometric features of environments, as LiDARs primarily provide range measurements instead of motion measurements. From now on, however, the situation changes thanks to the novel Frequency Modulated Continuous Wave (FMCW) LiDARs. FMCW LiDARs not only offer the point range with high resolution but also capture the instant point Doppler velocity through the Doppler effect. In the letter, we propose FMCW-LIO, a novel and robust LIO, leveraging intrinsic Doppler measurements from FMCW LiDARs. To correctly exploit Doppler velocities, a motion compensation method is designed, and a Doppler-aided observation model is applied for on-manifold state estimation. Then, dynamic points can be effectively removed by the Doppler criteria, deriving more consistent geometric observations. FMCW-LIO eventually achieves accurate state estimation and static mapping, even in structure-degenerated environments. Extensive experiments in diverse scenes are performed and FMCW-LIO outperforms other algorithms on both accuracy and robustness.