In 2009 we started, within the dedicated HARPS-Upgrade GTO program, an intense radial-velocity monitoring of a few nearby, slowly-rotating and quiet solar-type stars. The goal of this campaign is to gather, with high cadence and continuity, very-precise radial-velocity data in order to detect tiny signatures of very-low-mass stars potentially in the habitable zone of their parent stars. 10 stars have been selected among the most stable stars of the original HARPS high-precision program, uniformly spread in hour angle, such that three to four of them are observable at any time of the year. For each star we record 50 data points spread over the observing season. The data point consists of three nightly observations of a total integration time of 10 minutes each and separated by 2 hours. This is an observational strategy adopted to minimize stellar pulsation and granulation noise. In this paper we present the first results of this ambitious program. The radial-velocity data and the orbital parameters of five new and one confirmed low-mass planets around the stars HD 20794, HD 85512 and HD 192310, respectively, are reported and discussed, among which a system of three super-Earths and one harboring a 3.6 M⊕-planet at the inner edge of the habitable zone. This result already confirms previous indications that low-mass planets seem to be very frequent around solar-type stars and that this occurrence frequency may be higher than 30%.