  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. 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
{\AA}. 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 RC stars. 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), possibly
indicative of a high density peak in the central 250 pc of the bulge
