We searched for periodic and quasi-periodic signals in the prompt emission of a sample of 44 bright short gamma-ray bursts (GRBs) detected with Fermi/GBM, Swift/BAT, and CGRO/BATSE. The aim was to look for the observational signature of quasi-periodic jet precession, which is expected from black hole (BH)-neutron star (NS) mergers, but not from double NS systems. Thus, this kind of search holds the key to identifying the progenitor systems of short GRBs and, in the interim before gravitational wave detectors become on-lines, represents the only direct way to constrain the progenitors. We tailored our search to the nature of the expected signal by properly stretching the observed light curves by an increasing factor with time, after calibrating the technique with synthetic curves. None of our GRBs showed evidence for periodic or quasi-periodic signals. In particular, for the seven unambiguously short GRBs with the best signal-to-noise ratios, we obtained significant upper limits to the amplitude of the possible oscillations. This result suggests that BH-NS systems do not dominate the population of short GRB progenitors, as described by the kinematic model of Stone et al. (2013PhRvD..87h4053S).