If you stay south of Canton and Mansfield you should be fine -
Maybe connecting up with the Northern Tier at Monroeville on the Indiana border.
I've biked the backroads in central Ohio twice on tours - going diagonally, not E-W.
The Mohican Hills are really nice - and most backroads in central Ohio are very nice.
AAA puts out detailed maps of various sections of Ohio with all the backroads.
Call the Columbus office and ask.