  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 $\chi^2$ 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.
