Underwater sensor networks (UWSNs) are characterized by large energy consumption, limited power supply, low bit rate, and long propagation delay, as well as spatial-temporal uncertainty, which present both challenges and opportunities for media access control (MAC) protocol design. The time-division transmissions can effectively avoid collisions since different nodes transmit packets at different period of time. Nevertheless, in UWSNs with long propagation delay, in order to avoid collisions, the period of time is subject to be long enough, which results in poor channel utilization and low throughput. In view of the long and different propagation delay between a receiving node and multiple sending nodes in UWSNs, as long as there is no collision at the receiving node, multiple sending nodes can transmit packets simultaneously. Therefore, in this paper, we propose a MAC protocol of concurrent scheduling based on spatial-temporal uncertainty called CSSTU-MAC (concurrent scheduling based on spatial-temporal uncertainty MAC) for UWSNs. The CSSTU-MAC protocol utilizes the characteristics of temporal-spatial uncertainty as well as long propagation delay in UWSNs to achieve concurrent transmission and collision avoidance. Simulation results show that the CSSTU-MAC protocol outperforms the existing MAC protocol with time-division transmissions in terms of average energy consumption and network throughput.