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The underwater acoustic channel is fundamentally different from the terrestrial wireless channel. Its unique characteristics, such as slow propagation speed and small bit rate distance product, present both challenges and opportunities for media access control (MAC) protocol design. In existing handshaking-based MAC protocols, each successful handshake only allows an initiating sender to transmit a single or multiple consecutive data packets to its intended receiver. In a long propagation delay environment, this unidirectional data transmission often results in extremely poor channel utilization due to the long waiting time for the handshake to complete. By exploiting the channel's unique characteristics, we present a novel approach based on concurrent, bidirectional data packet exchange to improve the data transmission efficiency. To further amortize the high latency overhead, we adopt a packet bursting idea, where a sender-receiver pair can exchange multiple rounds of bidirectional packet transmissions. Based on these strategies, we propose an asynchronous handshaking-based MAC protocol, which we call bidirectional-concurrent MAC with packet bursting (BiC-MAC). Via extensive simulations, we compare BiC-MAC against two representative unidirectional handshaking-based protocols, as well as several existing MAC protocols. We demonstrate that BiC-MAC can significantly increase channel utilization and offer performance gains in terms of both throughput and delay, while achieving a stable saturation throughput. Our study highlights the value of adopting bidirectional, concurrent transmission in underwater networks.