Hey, cool—I leave for the canyon in a couple of days! Not rim-to-rim, four days in backcountry on Tonto Plateau. Beautiful place.
I normally am strongly pro-boot. I just think the extra foot protection and ankle support is worth a little extra weight, especially with today's boots. There are a lot of very light boots out on the market today.
But let's be candid, the rim-to-rim route is a pretty well-trodden, well-maintained path. No bouldering, stream-wading, or blow down-clambering. So if you're a fit hiker, have solid ankles, and are carrying a very light load (I want to be careful here, as a basic safety load with water, food, and extra clothing still is important), I think a decent pair of trail runners will be perfectly fine. Salomon's XA Comp 5 GTX trail runners ($125) would be great. Or the Montrail Sabino ($100), which might be a little cooler because they don't have a Gore-Tex liner. Same can be said for the La Sportiva Wildcat ($100).
You might also choose from one of several mid-height shoes that combine the light weight of a trail runner with a bit more heft and support. I really like Oboz Yellowstone II Mids, for instance ($145). They're very comfortable and light, yet surprisingly rugged and supportive. Or the Merrell Moab Ventilator Mid ($95), a highly breathable mid-height boot purpose-built for warm, dry weather.
None of these need much break-in—just a few check-out hikes to ensure fit is good, and you're off.
Hope you have a great trip! Send us some photos.