NaxCoO2, the parent compound of the recently synthesized superconductor NaxCoO2∶yH2O, exhibits bulk antiferromagnetic order below ∼20 K for 0.75≤x≤0.9. We have performed neutron scattering experiments in which we observed Bragg reflections corresponding to A-type antiferromagnetic order in a Na0.82CoO2 single crystal and characterized the corresponding spin-wave dispersions. The spin waves exhibit a strongly energy-dependent linewidth. The in-plane and out-of-plane exchange constants resulting from a fit to a nearest-neighbor Heisenberg model are similar in magnitude, which is unexpected in view of the layered crystal structure of NaxCoO2. Possible implications of these observations are discussed.