I stay on the Yellowstone River just north of the Park for 4 or more months each year, normally from the beginning of June to early October, and am traipsing around the mountains there all summer long, as well as fishing in some of the hike-into places in the Park.
The ratio of Grizzly attacks on people compared to the number of people who visit the area each year is low -- I only know of 3 such attacks in our area in the fall of 2007 -- I haven't yet heard the statistics from this past year.
All of the information previously provided is good. However, I've found that the best way to insure your safety when visiting in our area is to always go with one or more people that you can outrun. I'm living proof of its effectiveness.
Oh, and afterwards send
to these persons' relatives.