Transverse deuteron spin relaxation measurements, employing Carr-Purcell-Meiboom-Gill (CP) sequences, have been used to determine the anisotropic elastic constants of a thermotropic main chain/side chain liquid crystal polymer (LCP) in the nematic phase. The observed relaxation rates, R-2(CP)(omega), exhibit a square root dependence on the inverse pulse frequency, omega, i.e., R-2(CP)(omega)proportional toomega(-1/2), over more than one order of magnitude in omega in the kHz regime. This is precisely the dispersion law expected for nematic director fluctuations. Analysis of the experimental dispersion profile is performed using a slow-motional model for director fluctuations, in which five independent Leslie viscosities and three Frank elastic constants are considered. Using additional information from a step-rotation rheo-nuclear magnetic resonance (NMR) experiment, the analysis provides absolute values for the splay, bend, and twist elastic constant of the studied LCP. It is the first time that such data are available for this class of polymers. The splay elastic constant of K(1)similar to8x10(-8) N exceeds that of monomers by four orders of magnitude, in substantial agreement with theoretical predictions. The values for the bend and twist elastic constant of K(2)similar toK(3)similar or equal to5x10(-10) N are by a factor of 100 larger than those of low molecular weight liquid crystals. The results show that transverse NMR relaxation measurements involving CP sequences represent a powerful tool for the study of the anisotropic viscoelastic properties of LCPs. (C) 2003 American Institute of Physics.