H-1 spin lattice relaxation rate (R-1) dispersions were acquired by field-cycling (FC) NMR relaxometry between 0.01 and 35 MHz over a wide temperature range on polyisoprene (IR), polybutadiene (BR), and poly(styrene-co-butadiene) (SBR) rubbers, obtained by vulcanization under different conditions, and on the corresponding uncured elastomers. By exploiting the frequency-temperature superposition principle, chi"(omega tau(s)) master curves were constructed by shifting the total FC NMR susceptibility, chi ''(omega) = omega R-1(omega), curves along the frequency axis by the correlation times for glassy dynamics, tau(s). Longer tau(s) values and, correspondingly, higher glass transition temperatures were determined for the sulfur-cured elastomers with respect to the uncured ones, which increased by increasing the cross-link density, whereas no significant changes were found for fragility. The contribution of polymer dynamics, chi(pol)"(omega), to chi"(omega) was singled out by subtracting the contribution of glassy dynamics, chi(glass)''(omega), well represented using a Cole-Davidson spectral density. For all elastomers, chi(pol)"(omega) was found to represent a small fraction, on the order of 0.05-0.14, of the total chi"(omega), which did not show a significant dependence on cross-link density. In the investigated temperature and frequency ranges, polymer dynamics was found to encompass regimes I (Rouse dynamics) and II (constrained Rouse dynamics) of the tube reptation model for the uncured elastomers and only regime I for the vulcanized ones. This is clear evidence that chemical cross-links impose constraints on chain dynamics on a larger space and time scale than free Rouse modes.