Aims: We want to study the effects of the formation of planets and planetary systems on the atmospheric Li abundance of planet host stars. Methods: In this work we present new determinations of lithium abundances for 326 main sequence stars with and without planets in the Teff range 5600-5900 K. The 277 stars come from the HARPS sample, the remaining targets were observed with a variety of high-resolution spectrographs. Results: We confirm significant differences in the Li distribution of solar twins (Teff = T⊙ ± 80 K, log g = log g⊙ ± 0.2 and [Fe/H] = [Fe/H]⊙ ± 0.2): the full sample of planet host stars (22) shows Li average values lower than "single" stars with no detected planets (60). If we focus on subsamples with narrower ranges in metallicity and age, we observe indications of a similar result though it is not so clear for some of the subsamples. Furthermore, we compare the observed spectra of several couples of stars with very similar parameters that show differences in Li abundances up to 1.6 dex. Therefore we show that neither age, mass, nor metallicity of a parent star is the only cause for enhanced Li depletion in solar analogues. Conclusions: We conclude that another variable must account for that difference and suggest that this could be the presence of planets that causes additional rotationally induced mixing in the external layers of planet host stars. Moreover, we find indications that the amount of depletion of Li in planet-host solar-type stars is higher when the planets are more massive than Jupiter. Based on observations collected at the La Silla Observatory, ESO (Chile), with the HARPS spectrograph at the 3.6 m ESO telescope, with CORALIE spectrograph at the 1.2 m Euler Swiss telescope and with the FEROS spectrograph at the 1.52 m ESO telescope; at the Paranal Observatory, ESO (Chile), using the UVES spectrograph at the VLT/UT2 Kueyen telescope, and with the FIES, SARG, and UES spectrographs at the 2.5 m NOT, the 3.6 m TNG and the 4.2 WHT, respectively, operated on the island of La Palma in the Spanish Observatorio del Roque de los Muchachos.Table 6 is available in electronic form at http://www.aanda.org