Simple lattice model of polymer systems was developed and studied using the Monte Carlo method. The model chains were star-branched with f = 3 arms and rings. The number of polymer segments in a chain was varied up to 800. The chains were built on a simple cubic lattice with the excluded volume interactions only (the athermal system). The polymers were confined between two parallel impenetrable walls with a set of irregular obstacles what can be treated as porous media. A Metropolis-like sampling algorithm employing local changes of chain conformation was used. The dynamic properties of the model system were studied. The differences in the mobility of chains with different internal architectures were shown and discussed. The possible mechanisms of motions were presented.