Polymer coagulation is studied by a Monte Carlo diffusion model in which coagulant, solvent, and polymer particles move on nearest neighbor lattice sites according to the change in local interaction energy. Our approach has allowed, for the first time, to describe the coagulation process beyond the initial quench state and to reproduce the wide variety of different polymer structures that can be obtained, ranging from dust- to finger- to spongelike morphologies. We show that these morphologies are fully controlled by the coagulation rate which is itself strongly dependent on the degree of miscibility between solvent and coagulant.