We report on the detection of three Saturn-mass planets discovered with the HARPS instrument. HD 93083 shows radial-velocity (RV) variations best explained by the presence of a companion of 0.37 M_Jup orbiting in 143.6 days. HD 101930 b has an orbital period of 70.5 days and a minimum mass of 0.30 M_Jup. For HD 102117, we present the independent detection of a companion with m2sini = 0.14 M_Jup and orbital period P = 20.7 days. This planet was recently detected by Tinney et al. (2004). Activity and bisector indicators exclude any significant RV perturbations of stellar origin, reinforcing the planetary interpretation of the RV variations. The radial-velocity residuals around the Keplerian fits are 2.0, 1.8 and 0.9 m/s respectively, showing the unprecedented RV accuracy achieved with HARPS. A sample of stable stars observed with HARPS is also presented to illustrate the long-term precision of the instrument. All three stars are metal-rich, confirming the now well-established relation between planet occurrence and metallicity. The new planets are all in the Saturn-mass range, orbiting at moderate distance from their parent star, thereby occupying an area of the parameter space which seems difficult to populate according to planet formation theories. A systematic exploration of these regions will provide new constraints on formation scenarios in the near future.