Results of quasi-elastic neutron scattering (QENS) experiments on deuterated forms of pure poly(ethylene oxide) (PEO) and a PEO-based electrolyte are presented. Since hydrogen is an especially favorable element for exploiting QENS due to the considerable difference in coherent and incoherent scattering of hydrogen vs. deuterium, the results are compared to our previous data on hydrogenated materials with the same composition. The dynamics of the polymer, shown to be closely coupled to those of the Li ions, can be probed over a time scale ranging from pico- to nanoseconds by combining QENS and neutron spin-echo spectrometry (NSES). Finally, good agreement is found between the measured coherent and incoherent intermediate scattering functions for PEO and those obtained from molecular dynamics (MID) simulations.