Surface acoustic-wave phenomena span a wide range of length scales going from the devastation of earthquakes down to image reconstruction of buried nanostructures. In solid-fluid systems, the so-called Scholte-Stoneley waves (SSWs) dominate the scene at the interface with their evanescent fields decaying away into both media. Understanding and manipulating these waves in patterned surfaces would enable new applications of sound to be devised for imaging and acoustic signal processing, although this task has so far remained elusive. Here, we report SSW modes displaying directional gaps and band folding in fluid-loaded solid phononic plates. The plates are inhomogeneously patterned with in-plane periodic modulations of the elastic constants, but present flat surfaces free of corrugations. We experimentally demonstrate control of SSWs, which opens a promising route toward acoustic fluid sensing, microscopy, and signal processing.