We study the main (α-) and secondary (β-) relaxation in the plastically crystalline (PC) phase of cyanocyclohexane by various 2H nuclear magnetic resonance (NMR) methods (line-shape, spin-lattice relaxation, stimulated echo, and two-dimensional spectra) above and below the glass transition temperature T(g) = 134 K. Our results regarding the α-process demonstrate that molecular motion is not governed by the symmetry of the lattice. Rather it is similar to the one reported for structural glass formers and can be modeled by a reorientation proceeding via a distribution of small and large angular jumps. A solid-echo line-shape analysis regarding the β-process below T(g) yields again very similar results when compared to those of the structural glass formers ethanol and toluene. Hence we cannot confirm an intramolecular origin for the β-process in cyanocyclohexane. The fast β-process in the PC phase allows for the first time a detailed 2H NMR study of the process also at T > T(g): an additional minimum in the spin-lattice relaxation time reflecting the β-process is found. Furthermore the solid-echo spectra show a distinct deviation from the rigid limit Pake pattern, which allows a direct determination of the temperature dependent spatial restriction of the process. In Part II of this work, a quantitative analysis is carried out, where we demonstrate that within the model of a "wobbling in a cone" the mean cone angle increases above T(g) and the corresponding relaxation strength is compared to dielectric results.