
Bane Wraith |

For a quick answer, I'd recommend following the progression found on Improved Natural Attack ... But I wouldn't have a way of justifying this as correct by the rules.

Avianfoo |

how to get 3d8:
1) Take the tiny/large weapon damage table.
2) 2d6's large version becomes 3d6 (plain from the table)
3) but since there is no 3d6 entry, look at the 1d6 entry. That entry goes up to 1d8.
4) apply that 3 times and you get 3d6 -> 3d8.
Whether this is the correct or intended advancement is another story. The progression in the Improved Natural Attack looks closer to what is intended, but since there is no 3d6 entry in the tiny/large weapon damage table this is hard to verify.

Bane Wraith |

Looking at the same table... By that logic, wouldn't 2d8 turn into 4d6 instead of 3d8? Or 2d6 turn into 2d8?
...There doesn't seem to be a pattern there... I would not suggest looking for one where it isn't written. Improved Natural Weapon has one that is written, and that ALSO abides by everything that's on that table.
According to it, the next step in the progression is 4d6

Bane Wraith |

Again, the Improved Natural Attack feat seems to have a progression that both abides by the table and progresses beyond it.

Bart Vervaet |
I seem to remember reading somewhere long ago that a size increase roughly equals x1.5.
If possible, multiply the number of dice by 1.5 (so 2d6 would become 3d6)
If this is not possible multiply the dice size and round down (so 1d6 would become 1d8, and 1d8 would become 1d12, which is changed to 2d6).
Again, I do not have an official source for this, but somehow when I read this thread it popped into my mind, probably from somewhere in the 3.x days or even earlier.