  Context. Considerable effort has been put into using light curves observed by
space telescopes such as CoRoT, Kepler and TESS for determining stellar
rotation periods. While rotation periods of active stars can be reliably
determined, the light curves of many older and less active stars (e.g. stars
similar to the Sun) are quite irregular, which hampers determination of their
periods.
  Aims. We examine the factors causing the irregularities in stellar brightness
variations and develop a method for determining rotation periods of low
activity stars with irregular light curves.
  Methods. We extend the Spectral And Total Irradiance Reconstruction (SATIRE)
approach for modelling solar brightness variations to Sun-like stars. We
calculate the power spectra of stellar brightness variations for various
combinations of parameters defining the surface configuration and evolution of
stellar magnetic features.
  Results. The short lifetime of spots in comparison to the stellar rotation
period as well as the interplay between spot and facular contributions to
brightness variations of stars with near solar activity cause irregularities in
their light curves. The power spectra of such stars often lack a peak
associated with the rotation period. Nevertheless, the rotation period can
still be determined by measuring the period where the concavity of the power
spectrum plotted in the log-log scale changes sign, i.e. by identifying the
position of the inflection point.
  Conclusions. The inflection point of the (log-log) power spectrum is found to
be a new diagnostic for stellar rotation periods that is shown to work even in
cases where the power spectrum shows no peak at the rotation rate.
