Context. The space missions CoRoT and Kepler have provided photometric data of unprecedented quality for asteroseismology. A very rich oscillation pattern has been discovered for red giants, including mixed modes that are used to decipher the red giants' interiors. They carry information on the radiative core of red giant stars and bring strong constraints on stellar evolution. Aims. Since more than 15 000 red giant light curves have been observed by Kepler, we have developed a simple and efficient method for automatically characterizing the mixed-mode pattern and measuring the asymptotic period spacing. Methods. With the asymptotic expansion of the mixed modes, we have revealed the regularity of the gravity-mode pattern. The stretched periods were used to study the evenly space periods with a Fourier analysis and to measure the gravity period spacing, even when rotation severely complicates the oscillation spectra. Results. We automatically measured gravity period spacing for more than 6100 Kepler red giants. The results confirm and extend previous measurements made by semi-automated methods. We also unveil the mass and metallicity dependence of the relation between the frequency spacings and the period spacings for stars on the red giant branch. Conclusions. The delivery of thousands of period spacings combined with all other seismic and non-seismic information provides a new basis for detailed ensemble asteroseismology.