In the context of wave propagation through a three-dimensional acoustic medium, an analytical approach is developed to study the high-frequency multiple diffraction by thin rigid obstacles, including evaluation of the boundary edge effects. Starting from a multi-fold integral representation of Kirchhoff type, we apply suitable asymptotic estimates to the arising diffraction integrals so as to derive explicit formulas regarding two concrete examples of double reflection from flat surfaces. The acoustic ray-beam can change the plane of its propagation. The validity of such formulas is tested by comparison with the results from direct numerical treatments of the main integrals involved.