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Co2+ ions in an octahedral crystal field stabilize a jeff=1/2 ground state with an orbital degree of freedom and have been recently put forward for realizing Kitaev interactions, a prediction we have tested by investigating spin dynamics in two cobalt honeycomb lattice compounds, Na2Co2TeO6 and Na3Co2SbO6, using inelastic neutron scattering. We used linear spin wave theory to show that the magnetic spectra can be reproduced with a spin Hamiltonian including a dominant Kitaev nearest-neighbor interaction, weaker Heisenberg interactions up to the third neighbor, and bond-dependent off-diagonal exchange interactions. Beyond the Kitaev interaction that alone would induce a quantum spin liquid state, the presence of these additional couplings is responsible for the zigzag-type long-range magnetic ordering observed at low temperature in both compounds. These results provide evidence for the realization of Kitaev-type coupling in cobalt-based materials, despite hosting a weaker spin-orbit coupling than their 4d and 5d counterparts.