We developed the slowly relaxing local structure (SRLS) approach for analyzing NMR spin relaxation in proteins. SRLS accounts for dynamical coupling between the tumbling of the protein and the local motion of the probe and for general tensorial properties. It is the generalization of the traditional model-free (MF) method, which does not account for mode-coupling and treats only simple tensorial properties. SRLS is applied herein to 41 relaxation of (CDH2)-C-13 groups in the complex of Ca2+-calmodulin with the peptide smMLCKp. Literature data comprising H-2 T-1 and T-2 acquired at 14.1 and 17.6 T, and 288, 295, 308, and 320 K, are used. We find that mode-coupling is a small effect for methyl dynamics. On the other hand, general tensorial properties are important. In particular, it is important to allow for the asymmetry of the local spatial restrictions, which can be represented in SRLS by a rhombic local ordering tensor with components S-0(2) and S-2(2). The principal axes frame of this tensor is obviously different from the axial frames of the magnetic tensors. Here, we find that -0.2 <= S-0(2) <= 0.5 and -0.4 <= S-2(2) <= 0. MF features a single "generalized" order parameter, S, confined to the 0-0.316 range; the local geometry is inherently simple. The parameter S is inaccurate, having absorbed unaccounted for effects, notably Si-2(2) not equal 0. We find that the methionine methyls (the other methyl types) reorient with rates of 8.6 x 10(9) to 21.4 x 10(9) (0.67 x 10(9) to 6.5 x 10(9)) 1/s. The corresponding activation energies are 10 (10-27) kJ/mol. By contrast, MF yields inaccurate effective local motional correlation times, re, with nonphysical temperature dependence. Thus, the problematic S- and tau(e)-based MF picture of methyl dynamics has been replaced with an insightful physical picture based on a local ordering tensor related to structural features, and a local diffusion tensor that yields accurate activation energies.