Here's a Cuyahoga County map of 2010 turnout as a percentage of 2008 turnout by municipality, except for the City of Cleveland which is broken down into area that are some combination of 2008 wards, 2010 wards, and the neighborhood stistical areas. This is about the best I can do with changing precinct and ward boundaries in most of the municipalities in the county:
Clearly there is a huge difference between urban and suburban areas. The area with the most consistent turnout is tiny Cuyahoga Heights which had 94% of its '08 turnout in '10 (and Kasich managed to increase upon McCain's victory margin from 2 votes to 3). For the most part the wealthier areas tended to retain voters better. The notable exception here being the high turnout in Parma, a poorer Polish and Ukrainian ethnic western suburb.
Within Cleveland itself, the poorer areas generally voted less, as well as the racially diverse neighborhoods. The diverse (45% white, 35% Hispanic, 20% black) Clark-Fulton neighborhood saw the largest fall-off in turnout. There is no obvious correllation between race and relative turnout. In fact the neighborhood with the largest relative turnout within the city itself was the far southeastern (97% black) Lee-Miles neighborhood and the second largest was the far west side's (83% white) West Park/Kamm's Corners neighborhood.
Kasich also managed to win a precinct (19-Q) in the West Park neighborhood, something John McCain failed to do.
Thanks, awesome map!