You shouldn't have any problems. When you present your passport in customs control, they don't normally ask for other forms of ID like your driver's license or anything. If the passport is valid, has a picture that looks like you, there should be no delay.
The only problem could be at the ticket check-in counter. They might see that your surnames don't match. In that case, kindly direct them to the official notation in the back of your passport on the "Ammendments and Endorsements" page. Trust me, you won't be the first married woman to go through with her maiden name on her passport.
There shouldn't be any harm in carrying a copy of your marriage certificate with you if it makes you feel better.
Enjoy your trip!
.....Kelly