The adsorption of flexible polymers from dilute solution in good solvents at attractive walls is studied by Monte Carlo simulation of a coarse-grained off-lattice model, varying chain length N and the strength epsilon of a short-range attractive wall potential. Unlike many previous studies, no chain end (or other monomer) is held fixed near the wall. In qualitative agreement with previous work on different models, we find an adsorption transition at a critical strength of the wall potential, where the chain configuration changes from three-dimensional to quasi-two-dimensional in the limit of very long chains. The dynamics of the chains is studied both above and below this adsorption threshold epsilon(c). Time-dependent mean-square displacements of monomers and the chain center of gravity as well as the associated relaxation times are studied in detail. In the adsorbed phase, relaxation times are found to scale as tau proportional to N-zeff with z(eff) approximate to 2.65 +/- 0.1 and the lateral diffusion constant scales as D-N proportional to N--yeff with y(eff) approximate to 1.1. Also small-scale motions of monomers in different distances from the adsorbing wall are investigated, and a qualitative picture for the dynamics of these partially adsorbed chains is developed.