There are mounds from the maya all over Belize and Guatamala. As long as you can deal with jungle. Permission is easy to get from the landowners. There is a place in Teakettle village in Belize. The farmers would not mind having you dig as long as you handed them a $10 bill and promised to fill any holes.
Alternatively go to the airport in Belize City. People will offer you artifacts at amazing prices. But I do not know how you would get them out of the county. The risk was to high for me. I like my freedom.