Actually do see quite a few rattlesnakes, probably at least one a week. Usually come across them stretched across the trail when you least expect it–calls for some creative, rapid manuevering or a test of you bunny-hopping skills.
Over the 20 years of riding up there, I have seen several bears, but they usually just shuffle off into the trees or brush. My biggest concern is running into a mother with new cubs in the early summer.
Have never seen a mountain lion in the local mountains, but have seen a few tracks over the years. The State Wildlife people estimate the population in the Sandia and Manzano mountains, adjacent to the city, to be only 15-20 at most (covers a 50 by 20 mile area). Some friends who live about 1/4 mile from me found 2 lion cubs in their backyard peachtree last summer, but mama was nowhere to be seen.
Coyotes are commonplace–I have opened my garage in the morning to walk the dogs to find one (or more) standing in the street. They saunter around in broad daylight. I was within about 15 feet from one along a walking trail yesterday when walking the dogs at 6AM.