The principal components of zeolite Si-29 magnetic shielding tensors have been accurately measured and calculated for the first time. The experiments were performed at an ultrahigh magnetic field of 21.1 T in order to observe the small anisotropies of the Si-29 shielding interactions that arise for Si atoms in near-tetrahedral geometries. A robust two-dimensional (2D) chemical shift anisotropy (CSA) recoupling pulse sequence was employed that enables quasi-static powder patterns to be resolved according to the isotropic chemical shifts. For the zeolites Sigma-2 and ZSM-12, it is demonstrated that the Si-29 chemical shift (CS) tensor components measured by the recoupling experiment are in excellent agreement with those determined from spinning sidebands in slow magic-angle spinning (MAS) experiments. For the zeolite ZSM-5, the principal components of the Si-29 CS tensors of 15 of the 24 Si sites were measured using the 2D CSA recoupling experiment, a feat that would not be possible with a slow MAS experiment due to the complexity of the spectrum. A simple empirical relationship between the Si-29 CS tensors and local structural parameters could not be established. However, the Si-29 magnetic shielding tensors calculated using Hartree-Fock ab initio calculations on clusters derived from the crystal structures are in excellent agreement with the experimental results. The accuracy of the calculations is strongly dependent on the quality of the crystal structure used in the calculation, indicating that the Si-29 magnetic shielding interaction is extremely sensitive to the local structure around each Si atom. It is anticipated that the measurement and calculation of Si-29 shielding tensors could be incorporated into the "NMR crystallography" of zeolites and other related silicate materials, possibly being used for structure refinements that may lead to crystal structures with very accurate Si and O atomic coordinates.