I believe I may be doubly descended from this Robert Greening. I haven't quite puzzled out all the exact connections yet, but I know that my great-great-great-grandmother was a direct descendant of his. I also know that her daughter (my great-great-grandmother) married another Greening (he'd have been my great-great-grandfather) from the county where Robert Greening's descendants all lived (Monroe), so if he was not directly descended from Robert, he was descended from one of Robert's siblings.
I saw one vague record which listed the original American forefather of this family to have come from Ireland after 1740. Does anybody know anything more about this?