The intermolecular nuclear magnetic resonance (NMR) chemical shifts of Xe in nanochannels of various dimensions and shapes are considered. Predictions of the line shapes that result from anisotropic averaging of the shielding tensor for Xe atoms in various limiting cases of nanochannels are made, based on ab initio calculations of Xe-129 shielding surfaces. Variation of the line shapes with channel cross sections, Xe loading, and with temperature are predicted for channels having cross-sectional areas that do not permit two or more Xe centers to be located on the same cross-sectional plane. It is shown that Xe in effectively one-dimensional channels should be expected to exhibit signature line shapes in Xe-129 NMR spectra, provided that the cross-sectional dimensions of the channels are sufficiently small that Xe atoms do not pass each other during diffusion. These predictions are tested against experimental examples of anisotropic Xe line shapes in various cavities and nanotubes.