Oxygen abundances for a large sample of dwarf stars in the Galactic thin and thick disks are determined from a non-LTE analysis of the oxygen triplet lines at 777 nm. Kinematic criteria are employed to determine whether a star belongs to the thin or thick disk. Temperatures are obtained from photometric calibrations based on the infrared flux method and surface gravities from Hipparcos parallaxes and stellar evolution calculations. High resolution spectra from the HET and the 2.7 m telescopes at McDonald Observatory, and the VLTI-UVES archive are used. Metallicities are derived from relatively unblended Fe I and Fe II lines for which reliable laboratory gf values are available. Oxygen abundances are obtained from the triplet lines at 777 nm and a restricted non-LTE analysis, i.e. spectrum synthesis was performed with non-LTE level populations on an LTE atmospheric structure. We confirm previous studies that suggest higher oxygen abundances in the thick disk than in the thin disk although our preliminary results favor a smooth transition instead of two completely separate trends, i.e. thick disk stars with intermediate and relatively lower oxygen abundances are also found.