It would depend almost entirely on your elevation and location. TYPICAL March temps in say, Superior, AZ (2,800') might be 45-70 low/high. Flagstaff (7,000') is typically 20-50 in March. Remember that some of the mountains on the AZT are pretty high and you lose 2-5 degrees for every 1,000' of elevation gain and visa versa.
Snow would depend on the previous Winter's fall, and other weather is pretty random. Anything North of the Mogollon Rim is pretty fickle in Spring. I live in Flagstaff and have seen 70 degree temps and sun in March, as well as heavy snow.
In general, most things South of the Mogollon Rim should be cool and favorable for hiking, bearing in mind you may encounter some snow in the higher mountains. North of the Mogollon is anyone's guess in March, though you will almost certainly hike through at least some snow and a TON of mud.
A couple years ago I hiked from the bottom of Grand Canyon to the top of Mt. Humphreys (AZ high point) in early March using the AZT. It was 80+ in the Canyon, with single digit temps in the San Francisco peaks, and I was on snow shoes. So pretty major swings totally depending on elevation- These two points are only about 60 air miles apart.
If you need to make reliable plans that would be unlikely to be affected by a late season snow, I might suggest picking a section South of the Mogollon Rim, and expect moderate to potentially heavy snow anywhere above 7-8,000 feet most years in March.