An integral identity developed by Brenner (1964) is used in the “reversed” context to derive a fixed point iterative scheme for the surface tractions on a rigid particle of arbitrary shape submerged in Stokes flows. The iterative approach facilitates the solution of very large systems of equations and thus the employment of high resolution discretization schemes. The utility of the approach is demonstrated by illustrative computations of the tractions on regular polyhedra, with special emphasis on the results near the edges and corners where (integrable) singular behavior is expected to provide a stringent test for the numerical method. Excellent agreement between our numerical estimates for the exponent of these singularities with the analytical result (local 2-D analysis) were obtained. The emphasis in this work is on validation of the approach, but references to applications, such as fluidic self-assembly of semiconductor microstructures, are provided.