In this work, a structural model of a metamaterial in the form of a chain of spherical mass-in-mass particles is constructed. Depending on the ratio between the acoustic wavelength and the lattice period, such a chain can behave as a phononic crystal or as an acoustic metamaterial. Dynamical equations for the chain are derived in the continuum approximation, when the lattice period is much smaller than the wavelength (this limit corresponds to a model of an acoustic metamaterial). Analytical dependences of the coefficients of the derived equation on the microstructure parameters of the medium (particle sizes and parameters of interparticle interactions) are obtained. The dispersion properties of the metamaterial under consideration are studied. In particular, frequency stop bands are found for both longitudinal and rotational waves depending on the microstructure parameters of such a metamaterial. It is established that the dispersion equation for coupled transverse-rotational waves contains two complex terms, i.e., wave attenuation appears in the system, which is typical for metamaterials. It is shown that for some values of the microstructure parameters of the medium, in the long-wavelength region there are three bands of forbidden frequencies. The width and location of the bands depend on the microstructure parameters.