Context. To fully exploit the scientific potential of the Fermi mission for the physics of active galactic nuclei (AGN), we initiated the F-GAMMA programme. Between 2007 and 2015 the F-GAMMA was the prime provider of complementary multi-frequency monitoring in the radio regime. Aims. We quantify the radio variability of gamma-ray blazars. We investigate its dependence on source class and examine whether the radio variability is related to the gamma-ray loudness. Finally, we assess the validity of a putative correlation between the two bands. Methods. The F-GAMMA performed monthly monitoring of a sample of about 60 sources at up to twelve radio frequencies between 2.64 and 228.39 GHz. We perform a time series analysis on the first 2.5-yr data set to obtain variability parameters. A maximum likelihood analysis is used to assess the significance of a correlation between radio and gamma-ray fluxes. Results. We present light curves and spectra (coherent within ten days) obtained with the Effelsberg 100 m and IRAM 30 m telescopes. All sources are variable across all frequency bands with amplitudes increasing with frequency up to rest frame frequencies of around 60-80 GHz as expected by shock-in-jet models. Compared to flat-spectrum radio quasars (FSRQs), BL Lacertae objects (BL Lacs) show systematically lower variability amplitudes, brightness temperatures, and Doppler factors at lower frequencies, while the difference vanishes towards higher ones. The time scales appear similar for the two classes. The distribution of spectral indices appears flatter or more inverted at higher frequencies for BL Lacs. Evolving synchrotron self-absorbed components can naturally account for the observed spectral variability. We find that the Fermi-detected sources show larger variability amplitudes, brightness temperatures, and Doppler factors than non-detected ones. Flux densities at 86.2 and 142.3 GHz correlate with 1 GeV fluxes at a significance level better than 3 sigma, implying that gamma rays are produced very close to the mm-band emission region.