In this talk we present a fluctuating boundary integral method (FBIM) for overdamped Brownian Dynamics of 2D periodic suspensions of rigid particles of complex shape immersed in a Stokes fluid. We develop a novel approach for generating Brownian displacements that arise in response to the thermal fluctuations in the fluid. Our approach relies on a first-kind boundary integral formulation of a mobility problem in which a random surface velocity is prescribed on the particle surface, with zero mean and covariance proportional to the Green’s function for Stokes flow (Stokeslet). This approach yields an algorithm that scales linearly in the number of particles for both deterministic and stochastic dynamics, handles particles of complex shape, achieves high order of accuracy, and can be generalized to three dimensions and other boundary conditions. FBIM provides the key ingredient for time integration of the overdamped Langevin equations for Brownian suspensions of rigid particles. We demonstrate that FBIM obeys discrete fluctuation-dissipation balance by performing equilibrium BD simulations of suspensions of non-spherical particles using a random finite difference temporal integrator.