I have been away for a few days so am just now catching up. Take a look at ohios-windy-9
Also the 555 is a real nice road south of Zanesville down to the river.
That's a real nice link to routes that provide a great "taste" of SE Ohio roads. I ride that area every couple of weeks and I see a few roads I've missed.
Besides the roads mentioned, a bit further east and north SR83 and SR78(east of MConnelsville to the Ohio River) are very nice for riding with good pavement and sweeping curves. Much of the area near I-77 was strip-mined for coal in the '60s and 70's. Homes and farms were removed, the topsoil disappeared, heavy equipment destroyed the roads. It was pretty ugly for awhile. After lots of "reclamation" the land is green, with trees and grasslands, and the roads were rebuilt. Most farms have not come back because there is no topsoil, the trees are not yet big enough for logging, the coal is gone, and there are few local jobs so the population is sparse. Most of the land is still owned by the power company. As a result, there is little traffic, only occasional trucks to damage the road, few intersecting driveways, and very little gravel or dirt on the road - all advantages for motorcycle riding.
As JimJim mentioned earlier, most any road in that corner of the state will be entertaining, especially if you stay on the State Routes. Expect more truck traffic as you near the Ohio River, those are the areas with logging, fracking and manufacturing. Set your waypoints carefully and check your itinerary because I've had my GPS send me down many gravel roads, some of which turned into two-tracks (I'm actually getting more comfortable on gravel - maybe I'll get out of 2nd gear someday.) If you have the time to plan ahead, a satellite view from Google usually makes the gravel apparent.
Personally, I do not
like the one road everybody talks about - State Route 555. Portions are narrow, with very tight curves and lots of sharp, blind hills. Many times you cannot tell where the road will go until you are within 10 feet of the top of the hill. Gravel on the pavement is common due to many driveways and vehicles dropping off the inside of the curves. Even at 35mph there is no time to see the scenery. This is the only road I've ever been on that made me feel nauseous while riding.
Enjoy your trip!