Vibration affects the service life of engineering structures and the working accuracy of instruments, causing engineering accidents such as structural collapse and equipment explosion. Phononic crystals can effectively suppress the propagation of elastic waves within the bandgap frequency range. In this study, based on the local resonance theory, a phononic crystal structure with zero Poisson’s ratio was designed by combining two classical structures: concave and chiral structures. The structure shows good zero Poisson’s ratio performance and a suppression effect on high-frequency vibration frequencies. Through numerical simulation and low-amplitude transmission tests, by changing the structure scale and material type, we realized an adjustable bandgap of the designed structure and confirmed the reliability of the results. Its structural design and research results provide new ideas for preventing high-frequency vibrations in industrial processes and provide reference values for the development of phononic crystals.