A significant challenge in autonomous driving systems lies in image understanding within complex environments, particularly dense traffic scenarios. An effective solution to this challenge involves removing the background or static objects from the scene, so as to enhance the detection of moving targets as key component of improving overall system performance. In this paper, we present an efficient algorithm for background removal in automotive radar applications, specifically utilizing a frequency-modulated continuous wave (FMCW) radar. Our proposed algorithm follows a three-step approach, encompassing radar signal preprocessing, three-dimensional (3D) ego-motion estimation, and notch filter-based background removal in the azimuth-elevation-Doppler domain. To begin, we model the received signal of the FMCW multiple-input multiple-output (MIMO) radar and develop a signal processing framework for extracting four-dimensional (4D) point clouds. Subsequently, we introduce a robust 3D ego-motion estimation algorithm that accurately estimates radar ego-motion speed, accounting for Doppler ambiguity, by processing the point clouds. Additionally, our algorithm leverages the relationship between Doppler velocity, azimuth angle, elevation angle, and radar ego-motion speed to identify the spectrum belonging to background clutter. Subsequently, we employ notch filters to effectively filter out the background clutter. The performance of our algorithm is evaluated using both simulated data and extensive experiments with real-world data. The results demonstrate its effectiveness in efficiently removing background clutter and enhacing perception within complex environments. By offering a fast and computationally efficient solution, our approach effectively addresses challenges posed by non-homogeneous environments and real-time processing requirements.