Aims: We seek to detect and quantify the effects of surface convection (granulation) on the line spectra of K-dwarfs as a first step towards a rigorous testing of hydrodynamic models for their atmospheres. Methods: Very high-resolution (R≃160 000-210 000), high signal-to-noise ratio (S/N⪆300) spectra of nine bright K-dwarfs were obtained with the 2dcoudé spectrograph on the 2.7 m telescope at McDonald Observatory to determine wavelength shifts and asymmetries of Fe i lines. Spectra of the same stars acquired with the High Resolution Spectrograph (R≃120,000) on the 9.2 m Hobby Eberly Telescope were used as radial velocity templates to calibrate the wavelength scale of the 2dcoudé spectra. Results: The observed shapes and positions of Fe i lines reveal asymmetries and wavelength shifts that indicate the presence of granulation. In particular, line bisectors show characteristic C-shapes while line core wavelengths are blueshifted by an amount that increases with decreasing equivalent width (EW). On average, Fe i line bisectors have a span that ranges from nearly 0 for the weakest lines (residual core flux ⪆0.7) to about 75 m s-1 for the strongest lines (residual core flux ≃0.3), while wavelength shifts range from about -150 m s-1 in the weakest (EW≃10 mÅ) lines to 0 in the strongest (EW⪆100 mÅ) features. A more detailed inspection of the bisectors and wavelength shifts reveals star-to-star differences that are likely associated with differences in stellar parameters, projected rotational velocity, and stellar activity. While the first two are understood and confirmed by our data, the relation to stellar activity, which is based on our finding that the largest departures from the expected behavior are seen in the most active stars, requires further investigation. For the inactive, slow projected rotational velocity stars, we detect, unequivocally, a plateau in the line-shifts at high EW values (EW⪆100 mÅ), a behavior that had been identified before only in the solar spectrum. The detection of this plateau allows us to determine the zero point of the convective blueshifts, which is useful to determine absolute radial velocities. Thus, we are able to measure such velocities with a mean uncertainty of about 60 m s-1 for four of our sample stars.