Medical image registration is crucial in medical image analysis applications. Recently, U-Net-style networks have been commonly used for unsupervised image registration, predicting dense displacement fields in full-resolution space. However, this process is resource-intensive and time-consuming for high-resolution volumetric image data. To address this challenge, this paper proposes a novel model named RegFSC-Net, which utilizes Fourier transform with spatial reorganization (SR) and channel refinement (CR) network for registration. We embed efficient feature extraction modules SR and CR modules into the encoder, and adopt a parameter-free model to drive the decoder to improve the U-shaped network. Precisely, RegFSC-Net does not directly predict the full-resolution displacement field in space but learns the low-dimensional representation of the displacement field in the bandlimited Fourier domain, which is beneficial in reducing network parameters, memory usage, and computational costs. Experimental results show that RegFSC-Net outperforms various state-of-the-art methods. Specifically, in comparison to the widely recognized Transformer-based method TransMorph, RegFSC-Net utilizes only around 8.2% of its parameters, resulting in a 1.95% higher Dice score and significantly faster inference speeds of 126.67% and 419.99% on GPU and CPU, respectively. Furthermore, we also designed three variants of RegFSC-Net and demonstrated their potential applications in computer-aided diagnosis.