The relaxation rate for the depolarization of a positive muon implanted in an antiferromagnetically coupled Heisenberg magnet is studied on the basis of a coupled mode theory of critical and paramagnetic spin fluctuations, which gives overall an unrivalled account of spin fluctuations. The paper includes the first comprehensive treatment of the dipole field that couples the muon and atomic (spin) magnetic moments. An analytical calculation of the relaxation rate is feasible in the vicinity of the critical temperature, because of the dominant role of the critical fluctuations, whereas in general numerical methods are required to calculate the spin response function and the dipole field. In the approach to the critical temperature, the muon relaxation rate increases as the square root of the correlation length. Results for the isotropic antiferromagnet RbMnF3 demonstrate that the relaxation rate is not a monotonic function of the temperature, and the magnitude and temperature variation of the relaxation rate depend to a significant degree on the site of the implanted muon. A theoretical framework for the interpretation of the muon relaxation rate is reviewed in a set of appendices.