The structure and the low-energy vibrational dynamics of CO and NO chemisorbed on a Rh(111) surface have been investigated by means of He-atom scattering (HAS). For a dilute phase of isolated CO molecules a characteristic frequency of (h) over bar omega =5.7 meV is observed which shifts to 5.45 meV for the (root 3x root3)R30 degrees phase (theta =1/3). Based on isotope exchange measurements this mode is assigned to the parallel frustrated translational mode (T-mode) of CO at on-top sites. For the (2x2)3CO saturation structure a further molecular vibrational mode with a frequency of (h) over bar omega =11.7 meV is obtained and is assigned to the T-mode of CO adsorbed at hollow sites which are predominantly populated in this phase. The He-atom diffraction patterns indicate a successive appearance of various NO superstructures upon increasing NO exposure including a low coverage c(4x2), an intermediate (3x3) and finally the (2x2)3NO saturation structure. The corresponding inelastic HAS measurements reveal again two characteristic molecular vibrational modes at 7.5 and 11.5 meV which are identified as the T-modes of NO adsorbed at on-top and hollow sites, respectively. Moreover, for the NO saturation phase two additional phonon modes appear at energies below 14 meV. Based on a lattice dynamical analysis of their dispersion curves they are identified as a back-folded Rayleigh mode and a further perpendicular polarized phonon mode caused by the reduced Brillouin zone of the NO adlayer. The different T-mode frequencies for CO or NO at on-top and hollow sites suggest that this mode is a sensitive signature of adsorption sites rather than the commonly used internal stretch mode which had led to wrong adsorption site assignments.