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With respect to other routing paradigms, source routing has received comparatively less attention in the underwater acoustic networking domain. The most likely causes of this lack of momentum are the high overhead caused by route discovery and maintenance in typical implementations of the source routing paradigm (e.g., dynamic source routing) in terrestrial radio networks. In this paper. we revert this view and argue that source routing can in fact be a reliable and convenient routing paradigm in underwater networks, when properly implemented and tailored to the peculiarities of underwater acoustic channels. Our scheme, named SUN, successfully recasts the source routing approach by introducing a number of new features, which improve the routing performance especially in the presence of unstable network links and mobile nodes. SUN is scenario-independent by design: this means that it can work in any connected topology, and does not need any side information (such as the node location and depth, or the channel state) in order to operate correctly. We evaluate the performance of SUN by means of simulations using the DESERT Underwater framework. Our results show that SUN correctly manages routing in both static and mobile networks, and that in some scenarios it even achieves better performance than a competing flooding-based approach. We also test the performance of SUN in a thorough experimental campaign involving six nodes and carried out in a lake near Berlin. From these results, we conclude that SUN, and the source routing paradigm in general, are in fact feasible options for general-purpose routing in underwater acoustic networks.