We study the Johari-Goldstein beta process of organic glass formers by one- (1D) and two-dimensional (2D) H-2 nuclear magnetic resonance (NMR). In particular, we compare systems with pronounced secondary relaxation in dielectric spectroscopy, namely toluene-d(5) and polybutadiene-d(6) (PB), with compounds which do not exhibit a distinct beta peak, i.e., glycerol-d(5) and polystyrene-d(3) (PS). Choosing large interpulse delays in the applied echo pulse sequences we increase the sensitivity on small angle rotational jumps. This way, we are able to probe clearly the beta process of toluene and PB in the line shape of 1D H-2 NMR spectra and in the orientational correlation functions of 2D H-2 NMR in time domain which is not possible when using the conventional techniques. Below the glass transition temperature T-g, the secondary relaxation of both glass formers is caused by a highly restricted reorientation of essentially all molecules. Comparing our results with simulations we estimate that the reorientation of most toluene molecules and PB monomeric units is characterized by an amplitude chi < 10 degrees. This amplitude is approximately unchanged below T-g, but strongly increases above the glass transition. Closer investigating the 1D H-2 NMR line shape for large interpulse delays we moreover demonstrate that the reorientation involved in the beta process takes place step-by-step via many elementary rotational jumps. On the other hand, for glycerol and PS, hardly any effects are observed in 1D and 2D H-2 NMR experiments below T-g when applying comparable experimental parameters. We conclude that reorientations with an amplitude chi >1 degrees do not occur on a time scale of mus-ms for the majority of molecules in glassy glycerol and PS. (C) 2001 American Institute of Physics.