I climbed Granite in 2000 following what I recall being a fairly average winter for snow. I ascended via the Rosebud/Avalanche Lake approach and used stiff leather mountaineering boots, crampons and an axe. I used running shoes for the approach, but found the climbing gear to be useful during the climb above Avalanche Lake.
The gully leading from the lake to the Tempest/Granite saddle was hard packed snow and a moderate grade. Climbing it in the early AM without crampons and an axe would have been a chore. Once above the saddle the I went through a period of taking my crampons on and off as I crossed steep, snow-filled chutes on the summit ridge. Now that I am more experienced, my guess is that I would be satisfied to just kick steps in these snow sections and use my axe.
Having said all that, if I was climbing that route today under similar conditions, I would definitely bring stiff boots and an axe and probably still bring crampons. If you ascend the "froze to death" plateau route, however, the need for crampons may even be less.
A more current and very informative account of a July 2005 climb with pictures from some of my friends here in New Mexico can be found here:
They are very skilled and experienced mountaineers and I think that its fair to say from their account that an axe is the minimum gear needed for an early summer ascent of Montana's highest.