Context. Planetary companions of a fixed mass induce larger amplitude reflex motions around lower-mass stars, which helps make M dwarfs excellent targets for extra-solar planet searches. State of the art velocimeters with $\sim$1m/s stability can detect very low-mass planets out to the habitable zone of these stars. Low-mass, small, planets are abundant around M dwarfs, and most known potentially habitable planets orbit one of these cool stars. Aims. Our M-dwarf radial velocity monitoring with HARPS on the ESO 3.6m telescope at La Silla observatory makes a major contribution to this sample. Methods. We present here dense radial velocity (RV) time series for three M dwarfs observed over $\sim5$ years: GJ 3293 (0.42M$_\odot$), GJ 3341 (0.47M$_\odot$), and GJ 3543 (0.45M$_\odot$). We extract those RVs through minimum $��^2$ matching of each spectrum against a high S/N ratio stack of all observed spectra for the same star. We then vet potential orbital signals against several stellar activity indicators, to disentangle the Keplerian variations induced by planets from the spurious signals which result from rotational modulation of stellar surface inhomogeneities and from activity cycles. Results. Two Neptune-mass planets - $msin(i)=1.4\pm0.1$ and $1.3\pm0.1M_{nept}$ - orbit GJ 3293 with periods $P=30.60\pm0.02$ d and $P=123.98\pm0.38$ d, possibly together with a super-Earth - $msin(i)\sim7.9\pm1.4M_\oplus$ - with period $P=48.14\pm0.12\;d$. A super-Earth - $msin(i)\sim6.1M_\oplus$ - orbits GJ 3341 with $P=14.207\pm0.007\;d$. The RV variations of GJ 3543, on the other hand, reflect its stellar activity rather than planetary signals.

Accepted for publication in A&A, 19 pages, 12 figures, 7 tables