We calculate the average power density spectra (PDSs) of 244 long gamma-ray bursts detected with the Swift Burst Alert Telescope (BAT) in the 15–150 keV band from 2005 January to 2011 August. For the first time we derived the average PDS in the source-rest frame of 97 gamma-ray bursts (GRBs) with known redshift. For 49 of them an average PDS was also obtained in a common source-frame energy band to account for the dependence of time profiles on energy. Previous results obtained on Burst and Transient Source Experiment (BATSE) GRBs with unknown redshift showed that the average spectrum in the 25–2000 keV band could be modelled with a power law with a 5/3 index over nearly two decades of frequency with a break at ∼1 Hz. Depending on the normalization and on the subset of GRBs considered, our results show analogous to steeper slopes (between 1.7 and 2.0) of the power law. However, no clear evidence for the break at ∼1 Hz was found, although the softer energy band of BAT compared with BATSE might account for that. We instead find a break at lower frequency corresponding to a typical source-rest-frame characteristic time of a few seconds. We furthermore find no significant differences between observer- and source-rest frames. Notably, no distinctive PDS features are found for GRBs with different intrinsic properties of the prompt emission either. Finally, the average PDS of GRBs at higher redshifts shows possibly shallower power-law indices than that of low-z GRBs. It is not clear whether this is due to an evolution with z of the average PDS.