© 2015 American Physical Society. Motivated by the recent heat transport experiments in two-dimensional antiferromagnets, such as La2CuO4, where the exchange coupling J is larger than the Debye energy ΘD, we discuss different types of relaxation processes for magnon heat current with a particular focus on coupling to three-dimensional phonons. We study thermal conductivity by these in-plane magnetic excitations using two distinct techniques: Boltzmann formalism within the relaxation-time approximation and memory-function approach. Within these approaches, a close consideration is given to the scattering of magnons by both acoustic and optical branches of phonons. A remarkable accord between the two methods with regards to the asymptotic behavior of the effective relaxation rates is demonstrated. Additional scattering mechanisms, due to grain boundaries, impurities, and finite correlation length in the paramagnetic phase, are discussed and included in the calculations of the thermal conductivity κ(T). Again, we demonstrate a close similarity of the results from the two techniques of calculating κ(T). Our complementary approach strongly suggests that scattering from optical or zone-boundary phonons is important for magnon heat current relaxation in a high-temperature window of ΘD T J.