South America is a vast continent and it would be easier to enumerate the places that are dangerous than those that are safe. It is true that violent crimes are more common than elsewhere but the majority of the continent is safe.
Crime rarely occurs in rural areas and so, even in a country like Ecuador where violence is rampant in many areas of Quito and Guayaquil, you can be completely comfortable elsewhere. A great example is the Galapagos islands which live off tourism but also trails in the Andes and coastal towns. Anything at the border of Colombia should be avoided though because there are armed militants and people do get caught in the cross-fire.
Peru is extremely touristy and, while there are some parts of Lima to avoid, nearly every place of interest is safe and patrolled by a dedicated and friendly tourist police, at least from morning to sundown. You do have to keep your wits about because crime does occur and pick-pockets are common in tourist areas.
Bolivia in general is also very safe for tourists. People are friendly and even La Paz is safe at night and the rest of the country even more so. As always, it is best to avoid dark and desolate places and if you keep in major roads, transportation and near places of interests, your chances for encountering crime are low.
Really, if you intend to visit historic centers and do outdoors type activities such as hiking, rafting, etc, I would expect you to need no more precautions than anywhere else. Argentina and Chile are even safer.
While I have not been to the rest of South America, I would still avoid Columbia for now, possibly with the exception of Cartegena in the north. Kidnappings, of foreigners in particular, is much higher there than elsewhere.