You should apply for a new passport on an expedited basis before you travel out of the United States to prevent issues at the border.
You should renew your passport under your married name, to match your name on your green card. Hopefully you also applied for, and obtained a state issued ID and a social security card under your married name.
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
I find both of colleagues answers sufficiently insightful. You will not get trouble with immigration on this. Our immigration system is complex and cumbersome. Sometimes overly bureaucratic too. But not on this particular issue.