The characterization of the structural and dynamic properties of rubber networks is of fundamental importance in rubber science and technology to design materials with optimized mechanical properties. In this work, natural and isoprene rubber networks obtained by curing at three different temperatures (140, 150, and 170 °C) and three different sulfur contents (1, 2, and 3 phr) in the presence of a 3 phr accelerator were studied using a combination of low-field time-domain NMR (TD-NMR) techniques, including 1H multiple-quantum experiments for the measurement of residual dipolar couplings (Dres), the application of the Carr–Purcell–Meiboom–Gill pulse sequence for the measurement of the transverse magnetization decay and the extraction of 1H T2 relaxation times, and the use of field cycling NMR relaxometry for the determination of T1 relaxation times. The microscopic properties determined by TD-NMR experiments were discussed in comparison with the macroscopic properties obtained using equilibrium swelling, moving die rheometer, and calorimetric techniques. The obtained correlations between NMR observables, crosslink density values, maximum torque values, and glass transition temperatures provided insights into the effects of the vulcanization temperature and accelerator/sulfur ratio on the structure of the polymer networks, as well as on the effects of crosslinking on the segmental dynamics of elastomers. Dres and T2 were found to show linear correlations with the crosslink density determined by equilibrium swelling, while T1 depends on the local dynamics of polymer segments related to the glass transition, which is also affected by chemical modifications of the polymer chains occurring during vulcanization.