A rigorous theory of the diffraction of SH-waves by a stress-free crack embedded in a semi-infinite elastic medium is presented. The incident time-harmonic SH-wave is taken to be either a uniform plane wave or a cylindrical wave originating from a surface line-source. The resulting boundary-value problem for the unknown jump in the particle displacement across the crack is solved by employing an integral equation approach. The unknown quantity is expanded in a complete sequence of Chebyshev polynomials. By writing the Green function as a Fourier integral, an infinite system of linear, algebraic equations for the expansion coefficients is obtained. Numerical results are presented for the particle displacement at the surface of the half-space, the far field radiation characteristic, the scattering cross-section of the crack and the dynamic stress intensity factor at the crack tips, for a range of geometrical parameters.