We present a new Monte Carlo lattice model to study the folding of two-dimensional heteropolymer chains. Previous lattice model studies of two-dimensional chains have been performed on square grids using predefined "move sets" of allowed moves. The motion of the polymer chain in these models is thus highly constrained. 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. From the results, we can characterize all of the possible one- and two-monomer moves and sort them according to relative importance. Greater flexibility in the lattice model produces several significant and interesting physical consequences.