© 2015 Elsevier Ltd. All rights reserved. Materials with negative elastic moduli are unstable, but can be stabilised by specific boundary conditions. In particulate materials under compression, rotating non-spherical particles produce the effect of negative Cosserat shear modulus. We consider wave propagation in such materials and demonstrate that when the sum of the negative Cosserat shear modulus and the conventional shear modulus is positive the waves can propagate. In the conventional isotropic Cosserat continuum the twist wave and one of the shear waves exist only at frequencies higher than a threshold. When the Cosserat shear modulus is negative all waves exist at all frequencies; observing the twist and shear waves one can detect and investigate the negative moduli.