We present and analyze a hybridized Nitsche method for acoustic metamaterials. The use of a stabilized primal-dual formulation allows us to cope with the sign-changing nature of the problem and to prove optimal error estimates under a well-posedness assumption. The method can be used on arbitrary shape-regular meshes (fitted to material interfaces) and yields optimal convergence rates for smooth solutions. As an illustration, the method is applied to simulate a realistic acoustic cloaking device.