Coby is onto something very simple and that very well can be the problem. Who installed your new clutch? Was the clutch kit specifically made for your car? The vibrations could be from multiple things, I personally believe it could be the detent springs on the clutch itself are rattling or whomever installed your clutch, did "not" set your clutch finger end gap correctly.
2009 Cobalt LS XFE U74
Mods: K&N Typhoon Intake, Option Cat-Back Exhaust, Cosmo Short shifter, Maxxim Ahead 17X7's, MPx shorty antenna, FE5 Strut/shocks, Tein S Techs, Moog endlinks, Moog spring seaters, ZZP shorty header, LSJ downpipe, LNF front grille