Aims: We present the results from a comprehensive spectroscopic survey of the WINGS (WIde-field Nearby Galaxy-cluster Survey) clusters, a program called WINGS-SPE. The WINGS-SPE sample consists of 48 clusters, 22 of which are in the southern sky and 26 in the north. The main goals of this spectroscopic survey are: (1) to study the dynamics and kinematics of the WINGS clusters and their constituent galaxies, (2) to explore the link between the spectral properties and the morphological evolution in different density environments and across a wide range of cluster X-ray luminosities and optical properties. Methods: Using multi-object fiber-fed spectrographs, we observed our sample of WINGS cluster galaxies at an intermediate resolution of 6-9 Å and, using a cross-correlation technique, we measured redshifts with a mean accuracy of ~45 km s-1. Results: We present redshift measurements for 6137 galaxies and their first analyses. Details of the spectroscopic observations are reported. The WINGS-SPE has ~30% overlap with previously published data sets, allowing us both to perform a complete comparison with the literature and to extend the catalogs. Conclusions: Using our redshifts, we calculate the velocity dispersion for all the clusters in the WINGS-SPE sample. We almost triple the number of member galaxies known in each cluster with respect to previous works. We also investigate the X-ray luminosity vs. velocity dispersion relation for our WINGS-SPE clusters, and find it to be consistent with the form Lx ∝ σ_v^4. Table 4, containing the complete redshift catalog, is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/495/707