I seen, heard and tried some of the "so-called" techniques offered here.
Bottom line is, a Dremel with jewelers rouge (The reddish-brownish stuff that Dremel offers in their polishing kits) will do the trick. Or you can acquire a stick of jewelers rouge from a watch supply house.
Use the hard felt polishing tip (make sure you use the FELT polishing tips), with plenty of jewelers rouge applied (turn on your Dremel and dip it in the container of polishing compound) and then with medium pressure keep the Dremel moving back and forth over the entire crystal in the direction of the scratch(es). IMPORTANT: KEEP THE DREMEL MOVING. DO NOT PAUSE OVER A SINGLE SCRATCH AND LEAVE IT RUNNING. THIS WILL CAUSE RIPPLES ON THE SURFACE OF THE CRYSTAL. It may take 15 or 20 minutes, or so, to get the scratches out. But it is much less messy that toothpaste and Brasso. And you are using a compound designed for crystal polishing. Leave Crest or Colgate to your teeth, and Brasso to your brass objects. THEN, when you have removed the scratches, switch to the soft, 1/2 inch round polishing pad (it's soft and pliable and comes in the Dremel kit). Apply ample amounts of the polishing compound and with medium to firm pressure go over the entire crystal to give it a mirror shine. Do this for about 5 minutes or so.
Since you are using jewelers rouge, which becomes a powdery substance on your Dremel polishing tips, I recommend WEARING A DUST MASK WHEN YOU ARE POLISHING YOUR CRYSTALS. 3M makes some inexpensive dust masks available from your friendly Wal-Mart.
I have removed some VERY DEEP scratches from crystals in this manner, with no mess and no ripple spots in the finished product. Takes a little longer, but it works. Always examine your work in progress with your loupe so as to see when the scratches are removed.
BTW, this jewelers rouge is great for polishing stainless steel watch cases, or casebacks to remove scratches. I DO NOT recommend using this on chrome or gold-tone cases.