Step-growth polymerization of rodlike molecules is qualitatively different from flexible molecules, because rotational and translational diffusion limitations result in a rate of reaction which is dependent on the molecular weights of the reacting oligomers. an understanding of the kinetics of polymerization is important for the manufacture of such polymers which have many applications. The theoretical basis and computation details of the pairwise Brownian dynamics method [Northrup er al., J. Chem. Phys. 80, 1517 (1984)] to determine the effective rate constant for reaction between rodlike molecules are presented. In this method the effective rate constant is obtained in terms of the first visit flux (for which an analytical expression is derived) and the reaction probability (which is obtained using Brownian dynamics simulations). A simple derivation for finite domain correction far the simulations is presented, which explicitly accounts for the spatial and orientational variations of the reaction probability. Computations are presented to validate assumptions of the technique. The computed rate constants decrease with decreasing rotational diffusivity to approach an asymptotic value, and decrease with decreasing values of the translational diffusivity perpendicular the rod axis. The computations show that the effective rate constants obtained for the case of dilute solutions when ratio of the translational diffusivities perpendicular and parallel to the rod axis is equal to 1/2, is close to the case of isotropic translational diffusion.