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 t2g manifold of iridates, and show that, while the Jahn-Teller effect does indeed not affect the jeff=1/2 antiferromagnetically ordered ground state, it leads to distinctive signatures in the jeff=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 jeff=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 jz. 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 Sr2IrO4.