Molecular dynamics (MD) simulations have been used to study diffusion of methane in three highly impermeable aromatic polyesters that are good barrier materials. These are amorphous poly(ethylene terephthalate) (PET) and poly(ethylene 2,6-naphthalene dicarboxylate) (PEN), and the nematic mesophase of the thermotropic liquid crystalline copolyester (LCP) of p-hydroxy benzoic acid (HBA) and 2,6 hydroxy naphthoic acid (HNA). Diffusion coefficients were determined in the temperature ranges of 450-625 K for PET, 500-625 for PEN, 425-530 K for the LCP, where values are large enough to be accessible to MD in practical computation times. Extrapolation, via Arrhenius plots, of the coefficients to near room temperature gave good agreement with experimental data in that region. This was found even though the glass transition temperatures (T-g) of PET (350 K) and PEN (390 K) lie in the intervening temperature range. This finding confirms previous observations that a low temperature hopping regime for small penetrants sets in on cooling well before the T-g of the host. Analysis of diffusant trajectories in terms of diffusive jump size distribution also shows that the low temperature hopping regime remains in place over the temperature range studied in these low diffusion coefficient polymers. Correlation of diffusion coefficients with free volume was examined. The LCP, even though diffusion there is highly anisotropic, is found to lie on a correlation found previously for five other polymers studied via MD. However, PET and PEN fail badly and are found to diffuse much more slowly than inferred from free volume vs diffusion coefficient behavior in the other polymers.