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The area of three-dimensional (3D) underwater acoustic sensor networks (UW-ASNs) have attracted significant attention recently from both academia and industry due to its applications in detecting and observing phenomena that cannot be adequately observed by means of two-dimensional ocean bottom underwater acoustic sensor nodes. Many problems arise with 3D UW-ASNs that need to be solved in order to enable underwater monitoring in the new environment. Among them, providing efficient routing is very challenging due to the unique characteristics of 3D UW-ASNs. In this paper, we present a Fault Tolerant Routing (FTR) protocol that addresses some of the major requirements imposed by 3D UW-ASNs including energy-efficient communication combined with high adaptability. We demonstrate through simulations that FTR achieves better performance than existing protocols in metrics of packet delivery ratio, end-to-end delay, network throughput and energy consumption.