Measurements conducted at the Spallation Neutron Source at ORNL have determined exchange couplings in the mag- netically ordered chalcogenide Fe1.05Te, a nonsuperconducting member of the iron-based family of superconductors. The results show the exchange pairings thought to be related to superconductivity occur in a next-nearest-neighbor ordering of atoms, rather than in nearest-neighbor order as in a previously studied iron arsenide.