[ABRIDGED] We determine Li abundances for a well-studied sample of 714 F and G dwarf, turn-off, and subgiant stars in the solar neighbourhood. The analysis is based on line synthesis of the Li line at 6707 {AA} in high-resolution and high signal-to-noise ratio echelle spectra, obtained with the MIKE, FEROS, SOFIN, UVES, and FIES spectrographs. The presented Li abundances are corrected for non-LTE effects. Out of the sample of 714 stars we are able to determine Li abundances for 420 stars and upper limits on the Li abundance for another 121 stars. 18 of the stars with well-determined Li abundances are listed as exoplanet host stars. Our main finding is that there are no signatures of Li production in the thick disk, but the Li abundance for stars of the same effective temperature is independent of metallicity for stars that can be associated with the Galactic thick disk. Significant Li production is however seen in the thin disk, with a steady increase towards super-solar metallicities. At the highest metallicities, however, around [Fe/H]~ +0.3, we tentatively confirm the recent discovery that the Li abundances level out. We hence contradict the recent finding in other studies that found that Li is also produced in the thick disk. This is likely due to the chemically defined selection criteria those studies used to define their thick disk samples. Age criteria that we use here, produce a thick disk stellar sample that is much less contaminated by thin disk stars, and hence more reliable abundance trends. [ABRIDGED] A conclusion that can be drawn is that no significant Li production, relative to the primordial abundance, took place during the first few billion years of the Milky Way, an era coinciding with the formation and evolution of the thick disk. [ABRIDGED]
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