  We have performed inelastic neutron scattering (INS) experiments to
investigate the magnetic excitations in the weakly distorted
face-centered-cubic (fcc) iridate double perovskites La$_2$ZnIrO$_6$ and
La$_2$MgIrO$_6$, which are characterized by A-type antiferromagnetic ground
states. The powder inelastic neutron scattering data on these geometrically
frustrated $j_{\rm eff}=1/2$ Mott insulators provide clear evidence for gapped
spin wave excitations with very weak dispersion. The INS results and
thermodynamic data on these materials can be reproduced by conventional
Heisenberg-Ising models with significant uniaxial Ising anisotropy and sizeable
second-neighbor ferromagnetic interactions. Such a uniaxial Ising exchange
interaction is symmetry-forbidden on the ideal fcc lattice, so that it can only
arise from the weak crystal distortions away from the ideal fcc limit. This may
suggest that even weak distortions in $j_{\rm eff}=1/2$ Mott insulators might
lead to strong exchange anisotropies. More tantalizingly, however, we find an
alternative viable explanation of the INS results in terms of spin models with
a dominant Kitaev interaction. In contrast to the uniaxial Ising exchange, the
highly-directional Kitaev interaction is a type of exchange anisotropy which is
symmetry-allowed even on the ideal fcc lattice. The Kitaev model has a magnon
gap induced by quantum order-by-disorder, while weak anisotropies of the Kitaev
couplings generated by the symmetry-lowering due to lattice distortions, can
pin the order and enhance the magnon gap. Our findings highlight how even
conventional magnetic orders in heavy transition metal oxides may be driven by
highly-directional exchange interactions rooted in strong spin-orbit coupling.
