Molecular dynamics of a main chain thermotropic liquid crystal polymer in the smectic A phase has been investigated using multipulse dynamic nuclear magnetic resonance (NMR) techniques. Transverse deuteron spin relaxation times T-2E(CP) from quadrupole echo pulse trains (modified Carr-Purcell-Meiboom-Gill sequence) measured for deuterons in the aromatic rings of the mesogenic units are obtained as a function of pulse spacing tau sample orientation theta(n), and temperature. Just below the nematic-smectic A phase transition, the relaxation times exhibit a linear dispersion regime T-2E(CP) similar to tau(-1) consistent with smectic director fluctuations. At lower temperatures, the dispersion step gradually disappears, indicating that faster molecular motions are the dominant transverse relaxation process. The observed anisotropy in T-2E(CP), measured at short pulse spacings, approximately follows the (sin(4) theta(N))(-1) dependence expected for axial diffusion in a highly ordered medium. Analysis of the experiments is achieved employing a density operator treatment based on the stochastic Liouville equation. The intramolecular motion is identified with phenyl ring flips and is the fastest process studied, with correlation times varying from 10(-10) to 10(-7) s over the temperature range investigated. Intermolecular (individual molecule) dynamics are somewhat slower and have been interpreted as rotational diffusion in an orienting potential. The correlation times for intermolecular motion exhibit non-Arrhenius behavior approaching the glass transition, following a temperature dependence described by the Williams-Landel-Ferry equation over six orders of magnitude. This result indicates a strong coupling of the intermolecular motion to the glass transition process. The slowest motion affecting transverse deuteron spin relaxation is assigned to smectic director fluctuations or undulation waves. Analysis of the T-2E(CP) dispersion yields information concerning the viscoelastic properties of the polymer. At T=418 K, a splay elastic constant of K-1=2 X 10(-11)N has been estimated. Using the experimentally accessible value for the long wavelength cutoff of the elastic modes, the root mean square fluctuation (theta(0)(2))(1/2) of the director is calculated to be 4 degrees.