Spatiotemporal trajectories collected from GPS-enabled devices are of vital importance to many applications, such as urban planning and traffic analysis. Due to the privacy leakage concerns, many privacy-preserving trajectory publishing methods have been proposed. However, most of them could not strike a good balance between privacy protection and good data utility. In this paper, we propose DP-STTS, a differentially private spatiotemporal trajectory synthesizer with high data utility, which employs a model composed of a start spatiotemporal cube distribution and a 1-order Markov process. Specially, DP-STTS firstly discretizes the raw spatiotemporal trajectories into neighboring cubes, such that the model size is limited and the model's tolerance for noise could be enhanced. Then, a Markov process is utilized for the next location point picking. After adding noise under differential privacy (DP) to the model, synthetic trajectories that preserve essential spatial and temporal characteristics of the real trajectories are generated from the noisy model. Experiments on one real-life dataset demonstrate that DP-STTS provides good data utility. Our code is available at https://github.com/Etherious72/DP-STTS.