As others already posted, there aren't many left.
In Dayton, I would suggest Airway Billiards.
In Cincinnati you have Michaels Billiards.
If you go a little further south into Northern Kentucky (Florence) there is a little pool room called Kick Shot Billiards. I think they have 4 of the 9 foot diamond tables and maybe 8 or 10 bar tables.
If it matters to you, there is no smoking at Michaels but they do allow it at Kick Shots. Airway I am not sure.
Depending on what you are looking for, you should be able to find action in all 3 places.
Good luck!
On a side note, I think if a new pool room was opened in the right location, it could do pretty well in the Cincinnati / Northern KY area.