Molecular dynamics simulations of the orientational dynamics of water molecules confined in narrow carbon nanotubes and nanorings reveal that confinement leads to strong anisotropy in the orientational relaxation. The relaxation of the aligned dipole moments, occurring on a time scale of nanoseconds, is 3 orders of magnitude slower than that of bulk water. In contrast, the relaxation of the vector joining the two hydrogens is ten times faster compared to bulk, with a time scale of about 150 fs. The slow dipolar relaxation is mediated by the hopping of orientational defects, which are nucleated by the water molecules outside the tube, across the linear water chain.