Center-of-mass and single monomer motion in grafted chains comprising a strongly stretched polymer brush in thermal equilibrium are studied by large scale molecular dynamics and Monte Carlo simulations of a coarse-grained model. Good solvent conditions are assumed. Our findings seriously question earlier theoretical predictions about the relaxation described by Rouse dynamics of brush coatings. Thus, the correlation functions of parallel and perpendicular components of the mean distance of the center-of-mass from the grafting site, the squared gyration radius and end-to-end distance, are found to deviate strongly from a simple exponential decay. While the relaxation times extracted from the initial slope of the center-of-mass and the gyration radius correlation functions are roughly compatible with the theoretical prediction tau((i))(z) proportional to N-3, tau((i))(xy) proportional to N-2, the relaxation times extracted from the late exponential decay scale as tau((f))(z) proportional to tau((f))(xy) proportional to N-Delta with an effective exponent Delta approximate to 3.7 for polymers of length 64 <= N <= 256. Moreover, by studying the dynamics of individual monomers from the grafting site (i = 1) up to the free chain end (i = N), it is found that monomers near the middle (i = N/2 + 1) exhibit the slowest relaxation, although the range of their mean square displacements is clearly much smaller than that of the end monomers. Similar results are found also in the case of brushes formed from ring polymers.