Could be one or several of many different problems. Live wallpaper, a misbehaving app, a custom launcher that hasn't been updated for ICS or the GS3, or even too many widgets slowing you down.
This phone is a beast and really should be able to run through the paces with some widgets and/or a live wallpaper. That said, my first step would be to update all your apps. Maybe you have one or a couple that have corrected an issue with the GS3 and you haven't updated it yet.
Next I'd try either taking down the live wallpaper or custom launcher -- if you have either. After all of the above try rebooting the phone and see if that makes any difference.
If none of that helps with your sluggish UI then I'd say you may have a lemon. It happens, a friend of mine at work was showing me hers today. Her phone ran well, probably as good as any non-galaxy s3, but side-by-side next to mine there was a difference. She called up Verizon and they're sending her a new one.
GL