I have just returned to Australia after a month's (car) touring of South Africa and a short trip to Zimbabwe. Crime is endemic in these countries simply because most people are so incredibly poor that stealing is the simplest way for many (obviously not all) to survive. I would strongly advise you to not travel alone and preferably have at least 2 companions. Moving around at night without a car or taxi is not an option, even in upmarket suburbs.
On the other hand there is some spectacular scenery, the wildlife (not to be seen from a bike though!) is amazing and the vast majority of locals are very friendly and helpful.