Ali wrote:

Since I'm moving, and it's looking like moving whole rose bushes aren't
going to be practical, I wondered if it was okay to root a few cuttings to
take with me. Would they survive in this heat, and what sort of pieces
should I cut?

They're not particularly rare or anything like that. A New Dawn, Mix 'n
Match (I think) and a pink rugosa. All of them are at the ratty end of their
first flush. Any tips on taking a few pieces?

July is a great time to root cuttings. Look at the American Rose
Society website,http://www.ars.org and check out the articles about
propagation at http://www.ars.org/explore.cfm/propagation/

New Dawn is widely available. It might be easier to buy a new plant.
There is enough to do with moving....