The equations describing an elastic isotropic Cosserat continuum are presented. Solutions of these equations for the various modes of plane harmonic waves in an infinite medium are briefly discussed. The analysis then concentrates on the development and interpretation of the surface‐wave solution for straight crested waves on a Cosserat half‐space. It is found that a wave analogous to the Rayleigh wave of the classical elasticity theory exists, except that it is a dispersive wave in this theory. The phase velocity of the surface wave may either increase or decrease with frequency, depending on the relative magnitude of the micromaterial moduli.