Hocking Hills and Old Man's Cave - near Athens, OH - http://www.hockinghills.com/
Beautiful caves and hiking paths, pretty waterfalls. Good walking shoes recommended!
Ohio Statehouse - Columbus, OH - http://www.ohiochannel.org/your_stat..._operation.cfm
Great location with grand columns and sweeping staircases. Best to shoot on Saturdays and Sundays - the offices are closed, and there are very few people around. No permit needed as long as you are not shooting professionally.
Whetstone Park of Roses - 3901 N High ST, Clintonville, OH (suburb of Columbus) -
13 acres of gardens, full of flowers from spring to fall. Public park, very popular with wedding parties!