Stationary clutter can seriously degrade the performance of radar sensors. In the specific context of vital-sign monitoring, this deterioration can lead to the impossibility of tracking the desired motions. Pure linear-frequency-modulated continuous-wave (LFMCW) radars have arisen as an interesting solution to monitor vital signs, featuring both an increased phase-based range precision and an advantageous range-isolation capability. In this study, the impact of clutter on healthcare LFMCW radars is mathematically analysed and a Doppler high-pass filtering technique is proposed for its suppression. Simulation results are provided to highlight the key aspects of the derived mathematical framework and associated Doppler processing. Real experiments are also conducted to prove the validity of the devised clutter-mitigation procedure.