Can I rest floor joists on a ledger lying under the beams?
No. Your joists must bear on the beam directly, using hangers or actually resting on it. Even if you'd bolt the ledger up to the bottom of the beam you'd be relying on cross-grain strength. Not a good situation. 

As to your beam sizing question, that's for an engineer to answer (with drawings in hand). We don't have enough information about the structure as a whole to properly answer that. I can speculate, since 2x8 lumber isn't used for floor systems anymore, that even tripled you wouldn't meet most codes.