A boundary-element method is implemented to simulate the motion of two-dimensional particles with arbitrary shapes suspended in a viscous fluid under conditions of Stokes flow. Numerical discretization of an integral equation of the first kind for the particle surface traction results in a system of linear algebraic equations for the components of the traction over boundary elements distributed along the particle contours, as well as for the velocity of translation and the angular velocity of rotation of the particles about designated centers. The linear system is solved by the method of successive substitutions based on a physically motivated iterative method that involves the decomposition of the influence matrix into diagonal blocks consisting of physical particle clusters and then carrying out updates by matrix-vector multiplication using the inverses of the diagonal blocks. The iterations are found to converge as long as the minimum gap between the particles is larger than a threshold that depends on the particle shape and level of numerical discretization. To improve the numerical efficiency, the stiffness of the governing equations due to lubrication forces developing between intercepting particles is removed by preventing the particles from approaching one another to within less than a specified distance. Simulations of singly periodic suspensions of circular or elliptical particles in semi-infinite shear flow bounded by a plane wall and in pressure-driven flow in a channel bounded by two parallel plane walls are carried out for an extended period of time. The results confirm the occurrence of particle migration due to an effective hydrodynamic diffusivity and illustrate the dependence of the suspension effective viscosity and dynamics of the microstructure on the solid-phase areal fraction and particle aspect ratio.