Using Google Maps, with Bicycling as the transportation mode (August 2015). I was able to identify about 111 miles of dedicated bike paths and a few bike friendly roads that fairly closely parallel the Ohio River. Local area paths that have less than about 2 miles and not connecting to bike friendly roads near the river, are ignored in this answer. At this time I have personally only validated the first 3 miles on the Three Rivers Heritage Trail near Pittsburgh.
~ 3 miles; Three Rivers Heritage Trail
~ 8 miles; bike friendly roads on and near Neville island
~ 6 miles; paths and bike friendly roads
Wellsburg OH to Wheeling OH
~ 20 miles; Yankee trail & Wheeling Heritage Trail
More detail about the section above in What are the navigation considerations on the Ohio River between Pittsburgh PA.& Wheeling WV?
~ 2 miles; Glen Dale to Moundsville Rail-Trail
~ 11 miles; Ohio River Scenic Byway, bike friendly roads (with gaps of unclear friendliness)
~ 30 miles; Mostly bike friendly roads, in both Kentucky and Ohio (some well away from the river on the Ohio Side)
~5 miles bike friendly parts of the Ohio River Scenic Byway
~26 miles; Louisville Riverwalk & Louisville Loop, paths with some bike friendly roads
When the Louisville Loop trail ends at Farnsley-Moremen Landing, there are no more bike trail of significance paralleling the Ohio River. There are several bike friendly roads beginning at Brandenburgh KY, and continuing to Cairo IN where the Ohio joins the Mississippi River. These do form a continuous bike friendly (per Google) route roughly following the rivers path. Some of them near the river at cities and towns, but often are 5 or 10 miles from the river. Depending on river access some of these sections may viable for peddle and paddle outings.