Proton Field-Cycling (FC) nuclear magnetic resonance (NMR) relaxometry is applied over a wide frequency and temperature range to get insight into the dynamic processes occurring in the plastically crystalline phase of the two isomers cyanocyclohexane (CNCH) and isocyanocyclohexane. The spin-lattice relaxation rate, R1(ω), is measured in the 0.01–30 MHz frequency range and transformed into the susceptibility representation χNMR″ω=ωR1ω. Three relaxation processes are identified, namely, a main (α-) relaxation, a fast secondary (β-) relaxation, and a slow relaxation; they are very similar for the two isomers. Exploiting frequency–temperature superposition, master curves of χNMR″ωτ are constructed and analyzed for different processes. The α-relaxation displays a pronounced non-Lorentzian susceptibility with a temperature independent width parameter, and the correlation times display a non-Arrhenius temperature dependence—features indicating cooperative dynamics of the overall reorientation of the molecules. The β-relaxation shows high similarity with secondary relaxations in structural glasses. The extracted correlation times well agree with those reported by other techniques. A direct comparison of FC NMR and dielectric master curves for CNCH yields pronounced difference regarding the non-Lorentzian spectral shape as well as the relative relaxation strength of α- and β-relaxation. The correlation times of the slow relaxation follow an Arrhenius temperature dependence with a comparatively high activation energy. As the α-process involves liquid-like isotropic molecular reorientation, the slow process has to be attributed to vacancy diffusion, which modulates intermolecular dipole–dipole interactions, possibly accompanied by chair–chair interconversion of the cyclohexane ring. However, the low frequency relaxation features characteristic of vacancy diffusion cannot be detected due to experimental limitations.