A three-dimensional model for polymerization processes in nonequilibrium environments has been constructed as an extension of an earlier two-dimensional polymer-growth-Hamiltonian model [M. Vogt and R. Hernandez, J. Chem. Phys. 115, 1575 (2001)]. The extension to three dimensions will permit a direct comparison with experiments, and presently allows us to observe the effects of reduced conformational trapping as one scales from two to three dimensions. The three-dimensional model exhibits surprisingly similar equilibrium and dynamic behavior to the two-dimensional model with the exception that it grows large polymers which are more compact. The radius of gyration scales weaker with the polymer size in three dimensions as has been seen in other models. The effects of temperature ramping and jumps on the dynamics of cross-polymerization have also been examined and exhibit a switch over from linear to nonlinear response for larger perturbations.