Would there be any issues if I were to use white oak as posts and hickory or elm as beams and joists? I mean as far as the shrinking rates, checking, etc?
Nate

No, there's no issue. They may shrink at a different rate, but two timbers of the same species may shrink and check differently as well. I've found hickory to check a lot, to the point where I've had to re-cut the pieces. I don't have experience with elm, but I imagine that it could be hard to work with the interlocking grain.

Did they check too much before or after putting the pieces all together? And, what were you using the hickory for?

In that case, they were naturally curved hickory braces, and they split in half before the raising. So they were relatively thin and small. On other larger pieces I've used (8x8 & 8x10 I think), they mostly checked after raising. If you've got the wood, then I'd go for it.