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We calculate magnon dispersions and damping in the Kitaev-Heisenberg model with an off-diagonal exchange $\mathrm{\ensuremath{\Gamma}}$ and isotropic third-nearest-neighbor interact ion ${J}_{3}$ on a honeycomb lattice. This model is relevant to a description of the magnetic properties of iridium oxides $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ and ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$, and Ru-based materials such as $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. We use an unconventional parametrization of the spin-wave expansion, in which each Holstein-Primakoff boson is represented by two conjugate Hermitian operators. This approach gives us an advantage over the conventional one in identifying parameter regimes where calculations can be performed analytically. Focusing on the parameter regime with the zigzag spin pattern in the ground state that is consistent with experiments, we demonstrate that one such region is $\mathrm{\ensuremath{\Gamma}}=Kg0$, where $K$ is the Kitaev coupling. Within our approach, we are able to obtain explicit analytical expressions for magnon energies and eigenstates and go beyond the standard linear spin-wave theory approximation by calculating magnon damping and demonstrating its role in the dynamical structure factor. We show that the magnon damping effects in both Born and self-consistent approximations are very significant, underscoring the importance of nonlinear magnon coupling in interpreting broad features in the neutron-scattering spectra.