When strong spin-orbit coupling removes orbital degeneracy, it would at the same time appear to render the Jahn-Teller mechanism ineffective. We discuss such a situation, the t_{2g} manifold of iridates, and show that, while the Jahn-Teller effect does indeed not affect the j_{eff}=1/2 antiferromagnetically ordered ground state, it leads to distinctive signatures in the j_{eff}=3/2 spin-orbit exciton. It allows for a hopping of the spin-orbit exciton between the nearest-neighbor sites without producing defects in the j_{eff}=1/2 antiferromagnet. This arises because the lattice-driven Jahn-Teller mechanism only couples to the orbital degree of freedom but is not sensitive to the phase of the wave function that defines isospin j_{z}. This contrasts sharply with purely electronic propagation, which conserves isospin, and the presence of Jahn-Teller coupling can explain some of the peculiar features of measured resonant inelastic x-ray scattering spectra of Sr_{2}IrO_{4}.