Molecular dynamics simulations were employed to study the bound layer in polymer/graphene-based nanocomposites. We focused on the dynamic behavior of polar (poly(acrylic acid)) and nonpolar (polystyrene) linear chains, at the interface with graphene-based fillers, bearing different affinities with the polymers and different nanoroughness. Four temperatures were examined with a step of 50 K and the lowest one 100 K above the T-g of the corresponding bulk polymers. The dynamic response of the adsorbed layer exhibited an increased departure from that of the bulk, accompanied by a higher degree of dynamical heterogeneity, especially for the oxidized graphene systems. In the case of the nonoxidized analogues, an increased anisotropy of polymer diffusion parallel and perpendicular to the filler's plane was observed. Dynamics within the adsorbed layer exhibited Arrhenius-like characteristics in all the examined systems, in agreement with recent experimental studies. The combination of the results associated with backbone torsional motion and the chain desorption process revealed a strong dynamic transition near both the examined nanosheets, almost 100 K above the bulk T-g. However, in the adjacent-to-the-adsorbed layer, polymer dynamic characteristics approached the bulk behavior. It is therefore implied that, depending on its average size, a polymer chain may experience different dynamic regimes across its length near the interface region.