We designed and developed a simple model of polymer chains. Three types of model chains with different internal macromolecular architecture were studied: linear, star-branched with f=3 arms of equal length and rings. The chains consisted of identical united atoms (segments) and were restricted to a simple cubic lattice with the excluded volume interactions only (the athermal system). The macromolecules were confined between two parallel impenetrable walls with a set of irregular obstacles that can be treated as a crude model of porous media. The properties of the model studied were determined by the Monte Carlo simulations. A Metropolis-like sampling algorithm with local changes of chains' conformation was used. The short- and long-time dynamic properties of the model system were studied. The differences in the mobility of chains and their fragments for different internal polymer architectures were shown and discussed. The possible mechanisms of chain's motion were discussed.