Context. Circumstellar discs around massive stars could mediate the accretion onto the star from the infalling envelope, and could minimize the effects of radiation pressure. Despite such a crucial role, only a few convincing candidates have been provided for discs around deeply embedded O-type (proto)stars. Aims. In order to establish whether disc-mediated accretion is the formation mechanism for the most massive stars, we have searched for circumstellar, rotating discs around a limited sample of six luminous (>10 5 L ) young stellar objects. These objects were selected on the basis of their IR and radio properties in order to maximize the likelihood of association with disc+jet systems. Methods. We used ALMA with ∼0 . 2 resolution to observe a large number of molecular lines typical of hot molecular cores. In this paper we limit our analysis to two disc tracers (methyl cyanide, CH 3 CN, and its isotopologue, 13 CH 3 CN), and an outflow tracer (silicon monoxide, SiO). Results. We reveal many cores, although their number depends dramatically on the target. We focus on the cores that present prominent molecular line emission. In six of these a velocity gradient is seen across the core, three of which show evidence of Keplerian-like rotation. The SiO data reveal clear but poorly collimated bipolar outflow signatures towards two objects only. This can be explained if real jets are rare (perhaps short-lived) in very massive objects and/or if stellar multiplicity significantly affects the outflow structure. For all cores with velocity gradients, the velocity field is analysed through position-velocity plots to establish whether the gas is undergoing rotation with rot ∝ R −α , as expected for Keplerian-like discs. Conclusions. Our results suggest that in three objects we are observing rotation in circumstellar discs, with three more tentative cases, and one core where no evidence for rotation is found. In all cases but one, we find that the gas mass is less than the mass of any embedded O-type star, consistent with the (putative) discs undergoing Keplerian-like rotation. With the caveat of low number statistics, we conclude that the disc detection rate could be sensitive to the evolutionary stage of the young stellar object. In young, deeply embedded sources, the evidence for discs could be weak because of confusion with the surrounding envelope, while in the most evolved sources the molecular component of the disc could have already been dispersed. Only in those objects that are at an intermediate stage of the evolution would the molecular disc be sufficiently prominent and relatively less embedded to be detectable by mm/submm observations.