We present Rossiter-McLaughlin observations of the transiting super-Earth 55 Cnc e collected during six transit events between January 2012 and November 2013 with HARPS and HARPS-N. We detect no radial-velocity signal above 35 cm/s (3-sigma) and confine the stellar v sin i to 0.2 +/- 0.5 km/s. The star appears to be a very slow rotator, producing a very low amplitude Rossiter-McLaughlin effect. Given such a low amplitude, the Rossiter-McLaughlin effect of 55 Cnc e is undetected in our data, and any spin-orbit angle of the system remains possible. We also performed Doppler tomography and reach a similar conclusion. Our results offer a glimpse of the capacity of future instrumentation to study low amplitude Rossiter-McLaughlin effects produced by super-Earths.