The ability of lyotropic liquid crystals to form intricate structures on a range of length scales can be utilized for the synthesis of structurally complex inorganic materials, as well as in devices for controlled drug delivery. Here we employ magnetic resonance imaging (MRI) for non-invasive characterization of nano-, micro-, and millimeter scale structures in liquid crystals. The structure is mirrored in the translational and rotational motion of the water, which we assess by measuring spatially resolved self-diffusion tensors and 2H spectra. Our approach differs from previous works in that the MRI parameters are mapped with spatial resolution in all three dimensions, thus allowing for detailed studies of liquid crystals with complex millimeter-scale morphologies that are stable on the measurement time-scale of 10 hours. The 2H data conveys information on the nanometer-scale structure of the liquid crystalline phase, while the combination of diffusion and 2H data permits an estimate of the orientational distribution of micrometer-scale anisotropic domains. We study lamellar phases consisting of the nonionic surfactant C10E3 in 2H2O, and follow their structural equilibration after a temperature jump and the cessation of shear. Our experimental approach may be useful for detailed characterization of liquid crystalline materials with structures on multiple length scales, as well as for studying the mechanisms of phase transitions.