A formulation of elastodynamic diffraction problems for sinusoidally in time varying disturbances in a linearly elastic medium is presented. Starting with the elastodynamic reciprocity relation, an integral representation for the particle displacement is derived. In it, the particle displacement and the traction at the boundary of the obstacle occur. From the integral representation, an associated integral equation is obtained by letting the point of observation approach the boundary of the obstacle. The “obstacle” may be either a rigid body, a void, or a body with elastic properties differing from those of its environment, or a combination of these. The integral equation thus obtained is well-suited for numerical treatment, when obstacles up to a few wavelengths in maximum diameter are considered.