The presence of surface convection in K-type dwarfs is revealed in very high quality spectra of nine bright stars. The observed asymmetries and wavelength shifts of the Fe I absorption line profiles are mainly due to granulation. The bisectors of the strongest Fe I lines have a span of about 100 m/s and the central wavelengths of the weakest Fe I lines are shifted by up to -200 m/s. The blueshifts decrease for stronger Fe I lines, but they become independent of line strength for equivalent widths larger than about 100 mA. The detection of this "plateau" in the velocity shifts of the strongest Fe I lines is necessary to remove the non-negligible uncertainty introduced by granulation in the determination of absolute radial velocities. Line profiles computed using a 3D model atmosphere accurately reproduce the observations, with statistical tests showing an agreement at the 95 % confidence level, which validates the 3D model for spectroscopic studies of abundances and fundamental parameters of K-dwarfs. We find that 3D effects reduce the difference in the iron abundance determined separately from Fe II and Fe I lines, which is about 0.15 dex for 1D models, by two thirds, thus alleviating significantly the iron ionization imbalance problem in K-dwarfs. However, the 3D iron abundances from Fe I lines show a small dependence with excitation potential, similar to the 1D case, possibly due to non-LTE effects that have not been taken into account. We also find that the 3D correction to the effective temperatures of solar metallicity K-dwarfs derived with the infrared flux method is about +30 K. Finally, we show that the 3D spectrum synthesis of molecular bands greatly improves the agreement with the observational data compared to the 1D analysis, which overestimates the abundances derived from molecular features by a factor of 2.