Context. The Galactic bulge is a massive, old component of the Milky Way. It is known to host a bar, and it has recently been demonstrated to have a pronounced boxy/peanut structure in its outer region. Several independent studies suggest the presence of more than one stellar populations in the bulge, with different origins and a relative fraction changing across the bulge area. Aims: This is the first of a series of papers presenting the results of the Giraffe Inner Bulge Survey, carried out at the ESO-VLT with the multifibre spectrograph FLAMES. Spectra of ~5000 red clump giants in 24 bulge fields have been obtained at resolution R = 6500, in the infrared Calcium triplet wavelength region at ~8500 Å. They are used to derive radial velocities and metallicities, based on new calibration specifically devised for this project. Radial velocities for another ~1200 bulge red clump giants, obtained from similar archive data, have been added to the sample. Higher resolution spectra have been obtained for ~450 additional stars at latitude b = -3.5, with the aim of investigating chemical abundance patterns variations with longitude, across the inner bulge. In total we present here radial velocities for 6392 red clump stars. Methods: We present here the target selection criteria, observing strategy and the catalog with radial velocity measurements for all the target stars. Results: We derive a radial velocity, and velocity dispersion map of the Milky Way bulge, useful to be compared with similar maps of external bulges, and to infer the expected velocities and dispersion at any line of sight. The K-type giants kinematics is consistent with the cylindrical rotation pattern of M-giants from the BRAVA survey. Our sample enables to extend this result to latitude b = -2, closer to the Galactic plane than probed by previous surveys. Finally, we find strong evidence for a velocity dispersion peak at (0, -1) and (0, -2), possibily indicative of a high density peak in the central ~250 pc of the bulge. Based on observations taken with ESO telescopes at the La Silla Paranal Observatory under program IDs 187.B-909 and 089.B-0830.Full Table 2 is available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/562/A66