Time correlation functions in the Lebwohl-Lasher model of nematic liquid crystals are studied using theory and molecular dynamics simulations. In particular, the autocorrelation functions of angular momentum and nematic director fluctuations are calculated in the long-wavelength limit. The constitutive relations for the hydrodynamic currents are derived using a standard procedure based on non-negativity of the entropy production. The continuity equations are then linearized and solved to calculate the correlation functions. We find that the transverse angular momentum fluctuations are coupled to the director fluctuations and are both propagative. The propagative nature of the fluctuations suppresses the anticipated hydrodynamic long-time tails in the single-particle autocorrelation functions. The fluctuations in the isotropic phase are, however, diffusive, leading to t^{-d/2} long-time tails in d spatial dimensions. The Frank elastic constant measured using the time correlation functions are in good agreement with previously reported results.