A dynamic Monte Carlo algorithm has been employed to explore the cyclization of flexible linear chains without hydrodynamic interactions. The chains are represented by the bond fluctuation model in the semidilute and concentrated (or melt) regimes. We have obtained the decay rates of the cyclization process for different values of chain length, concentration, and capture radius of the reacting ends. In agreement with the theoretical predictions, the rate decays clearly show a short-time and a long-time behavior. The results are compared with quantitative theoretical results for both the short- and long-time regimes.