Time dependent Monte Carlo (TDMC) simulations are performed on a diamond lattice to determine the effect of surface properties/conditions on the growth of diamond thin films on flat and stepped diamond {001}(2X1):H surfaces under chemical vapor deposition conditions, The gas-surface interface consists of reactions of incoming gas-phase species, such as H-2 molecules and H and CH3 radicals with surface radical, pi-bond and step edge sites on the diamond {001}(2x1):H surface. The rates and probabilities of adsorption, abstraction, desorption, and incorporation reactions, as well as the reverse reactions, are explicitly calculated either via molecular dynamics or transition state theory methods, or taken from experimental measurements. The TDMC method allows all these reactions to occur simultaneously, though probabilistically, at each time step. The microscopic and macroscopic characteristics of the growing film are observed as functions of time. Diamond films of 10 similar to 100 layers are grown in the simulation and the observed growth rate (similar to 0.5 mu m/h at 1200 K) is in agreement with experimental results. The contributions to the activation energy of growth by specific processes such as H abstraction, CH3 adsorption and CH2 incorporation into the trough sites have been determined. The contributions to the activation energies by specific processes are not linearly additive, and the CH3 adsorption at step edges leads to enhanced growth at the edges.