We present a novel 3D odometry method that recovers the full motion of a vehicle only from a Doppler-capable range sensor. It leverages the radial velocities measured from the scene, estimating the sensor's velocity from a single scan. The vehicle's 3D motion, defined by its linear and angular velocities, is calculated taking into consideration its kinematic model which provides a constraint between the velocity measured at the sensor frame and the vehicle frame. Experiments carried out prove the viability of our single-sensor method compared to mounting an additional IMU. Our method provides the translation of the sensor, which cannot be reliably determined from an IMU, as well as its rotation. Its short-term accuracy and fast operation (~5ms) make it a proper candidate to supply the initialization to more complex localization algorithms or mapping pipelines. Not only does it reduce the error of the mapper, but it does so at a comparable level of accuracy as an IMU would. All without the need to mount and calibrate an extra sensor on the vehicle.