Ground Strap vs. Ground Cable
Cute picture above.....Grounding IS a factor in this issue; there has to be a complete circuit path for the galvanic corrosion to proceed; it is a simple electrical circuit that has to be completed. Anything that can increase resistance in the path will reduce it and anything that makes the path better will increase the action.
Yes, it will happen naturally with any ions in the coolant, and that is the reason that a zinc sacrificial piece in the AL rad can be used to protect the AL radiator and to keep the AL from corroding. You can see an example of this in the galvanic chart here; zinc is to the left of the AL alloys and so will become the metal that gets sacrificed rather than the AL.
Regularly testing and treating the coolant to prevent it from becoming acidic is a common practice in trucks to keep this from happening. There are test strips used for coolant testing.
http://www.rowantechnology.com/technical-resources/galvanic-series/
Not sure how to totally prevent a ground path through an all metal AL radiator as pointed out by 67dart273, but avoiding a ground strap directly TO the radiator IF it has metal tanks has merit IMO. Of course, a lot of AL cored tanks in newer cars have plastic tanks so the AL core gets electrically isolated in that that way, but with rads like the Champions with their all-metal AL construction, then it is an item to be aware of.
BTW, with older brass radiators, the same galvaninc corrision was taking place all these years but we never worried about it or were aware of it as never became an issue for a very good reason. That's because the iron was to the left of the brass in the galvanic series, and so the iron became the sacrificial element and the brass was the non-sacrificial (more 'noble') element. That relatively thick cast iron's surface became sacrificed and we just never had any issues since it is cast so thick (for good reasons, obviously!). The opposite it now true with AL radiators since they are so thin walled (so that they CAN be good radiators).