Huge Nordic music fan/AZN-pop nonfan here!
I don't think Pariisin Kevät (Finnish) is widely known in the US yet, but they've been my favorite foreign group for a good year or two now. My favorite of theirs is
Metsän Laidalla Puut probably followed by
Sytytä Valo.
I think Kent (Swedish) is definitely better known, but I've only gotten into their stuff in the past few months. My favorites from them are
Isis & Bast and
Sundance Kid.
I love the metal that comes from the Nordic nations, but a lot of those groups (Sabaton, Korpiklaani, etc.) sing in English and/or are already very popular here.
I would be ecstatic if there were more Saami vocalists. The only band I've found so far is Intrigue, and I know Korpiklaani did some Saami stuff when they were still Shamaani Duo as well as having some joiking on their newest album.