Development and deployment of underwater acoustic (UWA) sensor networks have emerged as an efficient solution for many applications related to marine environment e.g., scientific, industrial, and military operations. The main tasks to execute are remote sensing and monitoring. Grid topology with multihop relaying is very useful, since it allows a wide area coverage as well as long distance data transmission. In this paper, we investigate architectures where sensing and monitoring data are forwarded over multiple lines independently. We are interested in transmission schedules which maximize network throughput. We prove that an optimal schedule is necessarily per-node fair. We also derive the upper bound. Furthermore, we present a low-complexity algorithm to find schedules achieving the upper bound, regardless of the size of the network.