Timing analysis can be a powerful tool for shading light on the still obscure emission physics and geometry of the prompt emission of gamma-ray bursts (GRBs). Fourier power density spectra (PDS) characterise time series as stochastic processes and can be used to search for coherent pulsations and, more in general, to investigate the dominant variability timescales in astrophysical sources. Because of the limited duration and of the statistical properties involved, modelling the PDS of individual GRBs is challenging, and only average PDS of large samples have been discussed in the literature thus far. We aim at characterising the individual PDS of GRBs to describe their variability in terms of a stochastic process, to explore their variety, and to carry out for the first time a systematic search for periodic signals and for a link between PDS properties and other GRB observables. We present a Bayesian procedure which uses a Markov chain Monte Carlo technique and apply it to study the individual power density spectra of 215 bright long GRBs detected with the Swift Burst Alert Telescope in the 15-150keV band from January 2005 to May 2015. The PDS are modelled with a power-law either with or without a break. Two classes of GRBs emerge: with or without a unique dominant time scale. A comparison with active galactic nuclei (AGNs) reveals similar distributions of PDS slopes. Unexpectedly, GRBs with subsecond dominant timescales and duration longer than a few ten seconds in the source frame appear to be either very rare or altogether absent. Three GRBs are found with possible evidence for periodic signal at 3.0-3.2{sigma} (Gaussian) significance, corresponding to a multi-trial chance probability of ~1%. Thus, we found no compelling evidence for periodic signal in GRBs. The analogy between the PDS of GRBs and of AGNs could tentatively hint at similar stochastic processes that rule BH accretion across different BH mass scales and objects. In addition, we find evidence that short dominant timescales and duration are not completely independent of each other, in contrast with commonly accepted paradigms.