We use neutron scattering to show that spin waves in the iron chalcogenide Fe1.05Te display novel dispersion clearly different from both the first principles density functional calculations and recent observations in the related iron pnictide CaFe2As2. By fitting to a Heisenberg Hamiltonian, we find that although the nearest-neighbor exchange couplings in the two systems are quite different, their next-nearest-neighbor (NNN) couplings are similar. This suggests that superconductivity in the pnictides and chalcogenides share a common magnetic origin that is intimately associated with the NNN magnetic coupling between the irons.