To reduce the end-to-end delay in EH-WSNs (energy-harvesting wireless sensor networks), medium access control protocols using pipelined-forwarding have been introduced and studied. In real-life applications, there are several situations where it is difficult to harvest more energy than the energy consumed. Therefore, it is crucial to design a MAC protocol that allows nodes to efficiently relay data without exhausting the power in pipelined-forwarding multihop transmission. In this paper, we propose a PP-MAC (priority-based pipelined-forwarding MAC) protocol that determines the priority of relay nodes based on the residual power and energy-harvesting rate. The proposed protocol determines the probability of a node becoming a relay node based on the priority of the node and attempts to access the channel in a distributed manner. Furthermore, the PP-MAC protocol controls the sleep interval based on the power conditions of the nodes. It also minimizes the power exhaustion problem by controlling the sleep interval based on the priority of the nodes. The performance of the proposed PP-MAC was evaluated via computer simulation, and the results indicated that PP-MAC could improve the network lifetime by mitigating the power imbalance of nodes.