A microscopic statistical mechanical theory of vibrational energy relaxation rates for polyatomic solutes in simple solvents is presented. The theory is based on a model of a spherical solute present at infinite dilution in a fluid of spherical solvent particles, and the solute-solvent interaction potential depends on the vibrational coordinates of the solute. The theory is applied to study the experimentally observed anomalous density and temperature dependence of vibrational relaxation rates in supercritical fluids in the vicinity of the critical point. A quantitative comparison of the theory with experiment is presented, and the agreement is satisfactory.