Nowadays, high-speed sampling and transmission is a foremost challenge of radar system. In order to solve this problem, a compressive sensing approach is proposed for radar target signals in this study. Considering the block sparse structure of signals, the proposed method uses a simple measurement matrix to sample the signals and employ a Block Sparse Bayesian Learning (BSBL) algorithm to recover the signals. The classical BSBL algorithm is applicable to real signal, while radar signals are complex. Therefore, a Complex Block Sparse Bayesian Learning (CBSBL) is extended for the radar target signal reconstruction. Since the existed radar signal compressive sensing models do not take block structures in consideration, the signal reconstruction of proposed approach is more accurate and robust, and the simple measurement matrix leads to an easy implementation of hardware. The effectiveness of the proposed approach is demonstrated by numerical simulations.