Frequency-modulated continuous wave (FMCW) light detection and ranging (LiDAR) is an emerging 3D ranging technology that offers high sensitivity and ranging precision. Due to the limited bandwidth of digitizers and the speed limitations of beam steering using mechanical scanners, meter-scale FMCW LiDAR systems typically suffer from a low 3D frame rate, which greatly restricts their applications in real-time imaging of dynamic scenes. In this work, we report a high-speed FMCW based 3D imaging system, combining a grating for beam steering with a compressed time-frequency analysis approach for depth retrieval. We thoroughly investigate the localization accuracy and precision of our system both theoretically and experimentally. Finally, we demonstrate 3D imaging results of multiple static and moving objects, including a flexing human hand. The demonstrated technique achieves submillimeter localization accuracy over a tens-of-centimeter imaging range with an overall depth voxel acquisition rate of 7.6 MHz, enabling densely sampled 3D imaging at video rate.