Flores writes, "My native country automatically changed my last name on my passport when I applied for renewal (I had to disclose that I got married in the meantime) but I have never intended to change my name and wish to keep my maiden name. I also thought that if I change my mind the best time would be to do it when I naturalize, but now I have a situation such that my passport and permanent resident cards have different names. If I leave the US and try to reenter, will there be a problem?"
Well, the best situation to be in for airline travel and immigration is that you have consistent names on identification documents (driver's license, passport, frequent flier account, visa, green card, etc.) and accordingly, it would be best if you file USCIS Form I-90 for a new green card so that it has the same last name. That way you will have peace of mind not only while going through immigration control in the United States but also while checking in at the airport in a foreign location (employees at airline counters may decide to very rigid about it and not let you board a flight to the US because the names do not match).