Metamaterials, composed of artificial periodic microstructures, possess unusual properties such as negative Poissons ratio, negative Youngs modulus, negative effective mass density and/or other properties that are not found in nature materials. Recently, much work has been done on studying the deformation-driven tunable metamaterials, while little has considered deformation of the microstructures in the linearly elastic region. In this study, we present examples of the anti-tetrachiral auxetic structures that are used as deformation-driven tunable metamaterials and can be deformed in the linearly elastic region, leading to change of the band structures for wave propagation. Finite element simulations were validated by the experimental observations, and the results were found to be in good agreement. The proposed structures could be exploited to turn on or off the bandgaps directionally in the linear-elastic region and those appeared in specific directions according to the deformed directions. This study not only provides options of solutions for acoustic filtering, but also opens avenues for the design of acoustic switches.