Context. The shape and kinematics of the metal-poor stellar component of the Galactic bulge are still poorly characterized, and, therefore, the origin of this component is not yet strongly constrained. RR Lyrae stars in the bulge have been reported to be associated with the spheroidal, relatively metal-poor component. They offer a way to trace this component with precise distances and therefore the possibility to calculate orbits and minimize contaminations. While a few studies of RR Lyrae spectra with medium and/or high resolution are now available, none target stars in the Galactic bulge. Aims. We present here a spectroscopic determination of Fe and α-element abundances for RR Lyrae stars in the Galactic bulge, with the main goal of providing a benchmark to calibrate other metallicity indicators, appropriate for this specific stellar population. Methods. We analyzed FLAMES/GIRAFEE spectra of 78 RR Lyrae stars (60 ab-type and 18 c-type). We applied a full-spectrum fitting technique to obtain the spectroscopic metallicity and overall α-element abundance. Distances were derived by means of a period─luminosity─metallicity relation, and orbits were computed by combining the radial velocities derived in our study with the proper motions from Gaia DR3. Results. The resulting metallicities peak at [Fe/H]median = −1.34 ± 0.04 and −1.44 ± 0.08 dex for ab and c-types, respectively. The majority of the bulge RR Lyrae are metal-poor stars with relatively high α-element abundances around [α/Fe] ∼ 0.25 ± 0.03 dex. We used our spectroscopic measurements to test different methods for deriving metallicities based on photometry, which utilize Fourier parameters in the light curves of the RR Lyrae. The data suggest a possible correlation between the metallicity difference and the [α/Fe] ratio, which requires further investigation. There are some ab-type RR Lyrae that show metallicities higher than ─1 dex and low [α/Fe] values. We kinematically studied these stars and find a difference between three stars with similar [α/Fe] values and the main group, indicating that they may be slightly younger and correspond to the disk population.