Masking by Moving (MByM), provides robust and accurate radar odometry measurements through an exhaustive correlative search across discretised pose candidates. However, this dense search creates a significant computational bottleneck which hinders real-time performance when high-end GPUs are not available. Utilising the translational invariance of the Fourier Transform, in our approach, Fast Masking by Moving (f-MByM), we decouple the search for angle and translation. By maintaining end-to-end differentiability a neural network is used to mask scans and trained by supervising pose prediction directly. Training faster and with less memory, utilising a decoupled search allows f-MbyM to achieve significant run-time performance improvements on a CPU (168 %) and to run in real-time on embedded devices, in stark contrast to MbyM. Throughout, our approach remains accurate and competitive with the best radar odometry variants available in the literature – achieving an end-point drift of 2.01 % in translation and 6.3 deg /km on the Oxford Radar RobotCar Dataset.