Here's my take, keeping in mind that I don't like riding in traffic, and would gladly trade a longer route for a more heavily trafficked one.

Using the Heart of Ohio trail seems like a good idea. And there are quite a few trails in Columbus that can lead you to the northeast part of town where you can link to Heart of Ohio. Just be aware that the gap between Sunbury and the trail at Centerburg would naturally suggest route 36 as the connector - there is quite a bit of traffic and not a great shoulder, but the gap is less than 10 miles. There are county road alternatives if you want to avoid 36.

At Mt. Vernon I think I'd take Mansfield Road north to North Liberty Road to Devilbiss Road, which in turn would lead to Butler. These are paved county roads with no shoulders - but very low traffic flow. This route would transition from scattered residential near Mt. Vernon, to mixed farmlands and woods as you move north.

I've been using online sources to virtual scout road routes for a while, and have a couple of things that make it more effective, and may be useful to you when picking a route.

When trying to link 2 points I'll first check Google with 'terrain' turned on, in order to avoid unnecessary hills, and check for routes in river/creek valleys (often with less hills).

On the satellite view I'll scan the road to see how many cars I see, using the shadows from the phone poles as a rough guide to the time of day, can make an educated guess on what the traffic flow could be. Using more than 1 satellite view will give the most accurate idea (check Bing maps, or use Google Earth and switch to different satellite views).

The different satellite views on Google Earth can also be useful if the road goes through lots of trees, and is obscured on the Google map or Bing map view. Changing to a satellite view from a different time period can often give a fall or winter view, when the foliage is down and the roads are visible.

Zoom in on the satellite view and you can usually see if the road is paved or gravel (check for dark tar patches to indicate paved, or see if the road color/texture matches that of the gravel driveways leaving it). Also use the street view on Google as much as possible to see what the paving/shoulder conditions are.

Of course none of this is fool proof. Roads can get repaved, shoulders widened, traffic flow can change etc etc. But it gives you a better chance of making an educated guess. Also, try checking on crazyguywithabike.com and see if anyone has a journal about traveling in that area.

Good luck. Rural Ohio is a great place to ride for the most part.

Steve Z