The agent won't care but the problem is that if the name on the account does not match the name on the ticket, they won't give you the credit.
So I would say yes, sign up under your married name. Otherwise you will have to send them a copy of your marriage certificate to have it changed (I'm guessing most programs will want proof, like Mileage Plus did when I changed mine). It's not a big deal but just one more hassle you can avoid by signing up under your married name now.
But wait. You are saying you're not changing your name in your passport? You have to book all tickets in the exact name that's in the passport. So whatever you do, make sure that all three match - name in passport = name in ticket = name on frequent flyer account.