We present the first dynamical study of Blue Straggler Stars (BSSs) in three Galactic globular clusters, NGC\,3201, NGC\,5139 ($��$Cen), and NGC\,6218, based on medium-resolution spectroscopy (R 10000) obtained with IMACS. Our BSS candidate selection technique uses HST/ACS and ESO/WFI photometric data out to $>\!4.5\,r_c$. We use radial velocity measurements to discard non-members and achieve a success rate of $\sim93\%$, which yields a sample of 116 confirmed BSSs. Using the penalized pixel fitting method (pPXF) we measure the $v\sin(i)$ values of the sample BSSs and find their distribution functions peaked at slow velocities with a long tail towards fast velocities in each globular cluster. We find that the BSSs in NGC\,3201 and NGC\,6218 which show $v\sin(i)\!>\!50$ km s$^{-1}$ are all found in the central cluster regions, inside a projected $2\,r_c$, of their parent clusters. We find a similar result in $��$Cen for BSSs with $v\sin(i)\!>\!70$ km s$^{-1}$ which are all, except for two, concentrated inside $2\,r_c$. In all globular clusters we find rapidly rotating BSSs that have relatively high differential radial velocities which likely put them on hyperbolic orbits, suggestive of strong dynamical interactions in the past. We estimate that all the observed rapidly rotating BSSs are likely to form in their central cluster regions no longer than $\sim\!300$ Myr ago. Using dereddened $V\!-\!I$ colors of our photometric selection we show that blue BSSs in $��$Cen with $(V-I)_0$<0.25 mag show a significantly increased $v\sin(i)$ dispersion compared with their red counterparts and all other BSSs in our sample, therefore strongly implying that fast rotating BSSs in $��$Cen are preferentially bluer, i.e. more massive. This may indicate that this particular blue BSS population was formed in an unique formation event and/or through an unique mechanism.

16 pages, 12 figures, accepted in ApJ