Many areas of Ohio are prime Morel Territory! Here are a few:
Huston Woods State Park near Oxford, I have personally found many Morels here. Lots of good woods, but be aware: there will be other shroomers looking!
This note from Charles R.: Nelsonville, Ohio, off of US33.East on 33 from Columbus, towards Athens, there will be a college and fields on the right hand side of 33. The first entrance to the school and the fields are right past a yellow building called "LS". Over the railroad tracks and on the right and left fields there will be many morels, this place is well known in the area. If you continue on towards the school, Hocking College, it will loop around back to 33. There are other fields off to the left between the road and the river. I have seen people carrying out garbage bags full of morels. You just push the grass to the side and find morels. mostly Big yellows.
Also along 33 from Lancaster to Logan along the road and between the river is good.