Oxygen abundances for a large sample of dwarf and giant stars kinematically selected to be part of the Galactic thin and thick disks have been determined from a non-LTE analysis of the O I triplet lines at 777 nm. The abundance analysis was performed using the infrared flux method temperature scale, trigonometric surface gravities, and accurate atomic data. Within this framework, the ionization balance of iron lines could not be satisfied and so we adopted the iron abundances from Fe II lines only given that they are relatively less sensitive to changes in the atmospheric parameters. We show the resulting [O/Fe] vs. [Fe/H] relationship and briefly discuss its implications.