Rokugan Ninja can get the highest amount of sneak attack dice per round.
There are a couple different ways you can look at things. You can multiclass from class to class to prc to prc trying to make every level a level that grants +1d6 sneak attack damage. The most I've managed to wrangle in there is 14d6 at 20th level and there is no way any DM would let you play a character with that many classes.
Another way is to use Bayushi's Technique from the Rokugan Campaign Setting. It is a feat that can be taken up to 5 times that increases your sneak attack damage by 1d6. Maxing out this feat on a ninja 19 rogue 1 nets 16d6 sneak attack damage at 20th level and is much more logistically legal.
But the way to get the most sneak attack damage is to get the most attacks per round. So the best sneak attacker I've seen is a ninja 19/ranger 1 with improved two weapon fighting, greater two weapon fighting, expert tactician, power attack, cleave, and frenzied attack*. You have to be human to get this many feats, and it leaves no room for bayushi's technique to increase your sneak attack damage. But, with 4 attacks from base attack, 3 from greater two weapon fighting, 1 from expert tactician, 1 from frenzied attack, 1 from haste (ring/boots of speed), and hopefully 1 from cleave, you'll end up with 11 attacks at 10d6 sneak attack each.
*Frenzied attack is found in AEG's War sourcebook.
110d6 sneak attack is hard to top in one round. The 16d6 sneak attacker would need 7 attacks in a round to top that and he's too tapped out on feats to get that high.
Alternatively, the Fang of Lloth prestige class from Song and Silence presents an interesting idea. Once you gain enough levels in it to acquire 6 arms, Improved Multiweapon Fighting** will net you a minimum of 12 attacks. Add in expert tactician for 13 if you want combat reflexes as your prerequisite or frenzied attack if you want power attack, either will up it to 13. I doubt you'll be able to afford both. If somehow you managed to garner 15 base attack some how and still have 9 levels of the fang of lloth class you could take greater multiweapon fighting for a bare minimum of 18 attacks in a full round action. You won't have 10d6 sneak attack, but even 7d6 sneak attack will beat the 110d6 of the previous example if you can squeeze those feats in. 7d6 sneak attack at 18 attacks would be 126d6 btw. Add in 18 shortsword hits and it's 144d6, add in expert tactician, cleave, and frenzied attack and it's 150d6 per round. But I don't see anyway to do that without being epic level.
**Improved Multiweapon Fighting is in the Non-Epic level feats section of the Epic Level Handbook.