GS,
I can't quote the law that says that open carry is legal in Ohio...I think it may be the result of a prior judicial ruling on a concealed weapons case, ie the reason the law in that case was broken was because the weapon was concealed, not that the person had a weapon.
I can tell you that if you are considering open carry in Ohio there is a catch in the law you need to be aware of....that is that you cannot open carry in your car. As the law stands now in Ohio guns in cars must be locked away where they are not immediately accessible by the driver and transported unloaded. I'm sure there are more specific details if you read the actual text of the code, but that's the main point...
Mark