I know Canada, Switzerland, Netherland, and U.S. terriorites doesn't require visa if you have GC. Please note: US terriorites, that are part of U.S., include U.S. Virgin Islands, Guam, Puerto Rico, Diego Gracia, and some U.S. pacific islands.
If you are on any U.S. immigration status such as H1, F1, L1, B1, B2, etc. or you have GC, they, of course, allow you to enter U.S. terriorites without any problem ONLY IF
1) you fly/sail FROM U.S. mainland to any U.S terriority.
2) or you fly/sail from Alaska or Hawaii to any U.S. terriority.
3) or you fly/sail from any U.S. terriority to any other U.S. terriority such as from Puerto Rico to U.S. Virgin Islands.
*** they require you to show your passport and your immigration status. That's all. They won't let you to enter without your passport or immigration status.
How about the following countries:
1) Mexico (I know GC holders went to Mexico for holiday without visas. All they did is to show their green card and passport. correct?),
2) Caribbean countries such as Bahamas, British Virgin Islands, Trindad & Tabago, spanish terriorities, and French terriorites (I think they require either French visa, Spanish visa, and British visa, but not sure),
3) Central American countries and
4) South American countries?