Wave propagation in layered materials with free sliding layers is analysed in 2D using anisotropic Cosserat continuum formulation, which accounts for layer bending. It is found that the 2D Cosserat continuum supports propagation of waves of three types: usual longitudinal and transverse (shear) waves and a new one – the rotational wave. The velocities of the longitudinal and transverse waves strongly depend upon the direction of their propagation with respect to the layering, but exhibit only marginal dispersion. Opposite to this the rotational wave is strongly dispersive: its velocity increases proportionally to the wavelength, but marginally depends upon the propagation direction. A prominent feature of the rotational wave is its extremely high velocity that can be considerably greater than the velocity of longitudinal wave propagating in the material of the layers. This high velocity can be used to distinguish the rotational waves. Furthermore, this property allows detecting areas of sliding, which is important in both materials engineering for non-destructive control and in geoscience for monitoring the state of the Earth's crust.