Characteristic variability timescales in blazar γ-ray light curves can provide insights into the physical processes responsible for γ-ray variability. The power spectral density (PSD) is capable of revealing such timescales, which may appear as breaks or periodicities. Continuous-time autoregressive moving-average (CARMA) models can be used to accurately estimate a light curve's PSD. Through a light-curve simulation study, we develop a methodology to identify PSD breaks using CARMA models. Using this methodology, we study the γ-ray light curves of 13 bright blazars observed with the Fermi Large Area Telescope in the 0.1-300 GeV band over 9.5 yr. We present the blazar γ-ray PSDs, which provide evidence for low-frequency breaks on timescales ∼1 yr in four sources, and an additional high-frequency break on a timescale ∼9 days in one source.