The rim at Bryce is at around 9000 ft, so its high and it gets cold. I was there in january and it was snowy though mild and pleasant during the day. In March you're looking at 50s during the day and 30 ish at night. It ain't hot desert. Zion is much lower and I'm not sure about that, but expect it to be 15 degrees warmer or so...
40-68 as this chart shows, so bring some warm clothes with you.
March is pretty pleasant still it can get cold and even an unusual snow shower. Just bring layers with a weatherproof shell and you should be fine. I like it better than the heat of summer.
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If the weather is good March can be a great month to check out the park. The beginning of the leaves arriving is a great antidote for cabin fever. Most likely you will need a jacket at the least, it would be unlikely to be shorts and t-shirt weather.
If the weather is bad you will get rain more likely than snow. And rain in the canyon far from being a bad thing will give you gorgeous waterfalls off all the cliffs. One of my favorite things I've ever seen there.
But you do have to come prepared for coat weather as well, because it can get down right cold still.
March --- very unpredictable