Many marine scientific, industrial, and military applications may require the deployment of underwater acoustic sensor networks for sensing and monitoring. A grid topology with multihop relaying is useful for wide area coverage as well as long distance data transmission. We investigate network architectures where data originate at one end of the grid, and are forwarded along multiple lines. We are particularly interested in transmission schedules that maximize network throughput by exploiting propagation delay to allow multiple simultaneous transmissions. We show that an optimal schedule is necessarily per-node fair, and derive the upper bound on throughput. Furthermore, we present a low-complexity algorithm to find schedules achieving the upper bound, regardless of the size of the network.