The biological activity of protein molecules is central to life. It has been known for decades that this biological activity is dependent on the protein molecule achieving a particular geometric conformation. Simple lattice models have been developed to investigate the protein folding pathway since all-atom molecular dynamics simulations on the time scale of folding are beyond the current capabilities of computers. We present a new Monte Carlo lattice model to study the folding of heteropolymer chains. Previous lattice model studies of two-dimensional chains have been performed on square grids using pre-defined "move sets" of allowed moves. The motion of the polymer chain in these models is thus highly constrained. In order to add greater flexibility, we use a triangular lattice and allow the chain to choose its own moves. Physically unrealistic moves are prevented by including kinetic energy effects in the Metropolis algorithm. By looking at the results, we are able to characterize all of the possible one and two particle moves in the two-dimensional model and sort them according to relative importance. This information will be used to guide simplified molecular dynamics studies. We also find that the initial phase of the folding process is a rapid collapse to a relatively compact state which is entropically driven. The model has been extended to three dimensions. The increased difficulty of working in three-dimensions as well as the preliminary results will be discussed.