Surface acoustic wave (SAW) devices have found a wide variety of technical applications, including SAW filters, SAW resonators, microfluidic actuators, biosensors, flow measurement devices, and seismic wave shields. Stretchable/flexible electronic devices, such as sensory skins for robotics, structural health monitors, and wearable communication devices, have received considerable attention across different disciplines. Flexible SAW devices are essential building blocks for these applications, wherein piezoelectric films may need to be integrated with the compliant substrates. When piezoelectric films are much stiffer than soft substrates, SAWs are usually leaky and the devices incorporating them suffer from acoustic losses. In this study, the propagation of SAWs in a wrinkled bilayer system is investigated, and our analysis shows that non-leaky modes can be achieved by engineering stress patterns through surface wrinkles in the system. Our analysis also uncovers intriguing bandgaps (BGs) related to the SAWs in a wrinkled bilayer system; these are caused by periodic deformation patterns, which indicate that diverse wrinkling patterns could be used as metasurfaces for controlling the propagation of SAWs.