We are in the process of studying strained-layer growth of two-dimensional Lennard-Jones lattices. To do so, we have developed three techniques, based on the Monte Carlo method and molecular dynamics, of simulating atomistic crystal growth from the vapor phase. The Monte Carlo method efficiently simulates the effects of long time-scale processes on the growth of strained-layer systems, but omits the transient dynamics of particle adsorption. The second technique, using molecular dynamics, gives results suggesting that epitaxial growth of strained-layer systems can occur on the picosecond timescales. However, this technique cannot capture the influence of the long time-scale processes on the growth process. In view of the shortcomings of the previous two techniques, A hybrid technique incorporating both the Monte Carlo method and molecular dynamics, has been developed. In principle, this technique models the transient dynamics of adsorption as well as the long term evolution of the system. This technique, however, is limited by artifacts that may only be eliminated by use of unwarrented amounts of supercomputer time.