Gifts when visiting Danes??

Not sure about Danish tradition, but a small 'housewarming' type gift from here in NC should be appropriate. Why not take some BBQ sauce (if you are checking a bag) or a lovely book with pictures of the moutains and the coast of NC. Or some peanuts, or some handcrafted items- lace, doilies, etc. Or pick up some Moravian cookies or cheese straws. Or a bottle of NC wine (again if checking a bag)- there's even a NC winery that makes honey mead.

Great ideas Toni! (flowers wouldn't make it through customs.).

Thought of a few more- toasted pecans, molassas candy, peanut brittle, or other 'local' candy, even flower seeds (I'd go with things like fox glove, touch-me-nots, etc- you can get 'heritage/heirloom' seeds from Old Salem in Winston-Salem). Emboirdered 'things'- like dresser scarves, placemats or napkins, etc- might be able to get something like that at the state fair in Raleigh. You could look for something with the dogwood flower (NC state flower) or pine (state tree) designs.

Toni has some interesting suggestions, but don't bring flower seeds. I don't know the rules for Denmark, but it many countries in is illegal to bring in seeds without the appropriate certification. You didn't mention your relationship with these families and that could make a difference; however, when you are invited to a home for dinner in Denmark, the typical gift is either flowers or a box of nice chocolates. If you buy flowers, get some assistance from the person at the flower shop on what to buy. There are certain types of flowers that are not appropriate (e.g. flowers traditionally sent for funerals), so you want to make sure that you bring the right type of flowers.

When we visited family in 2010 we bought a bottle of french wine and 2 bottles of akvavit. Our family really appreciated the gift.
Marilyn

All over Germany it's standard to bring a bottle of wine or flowers for the host. Something from home is as good or better. We didn't visit any homes in Denmark, but I'd expect it's much the same.

People at Denmark like candy. At many places in Europe, butterscotch candy is not available. If butterscotch candy is not for sale at Denmark, I guess people there would like receiving butterscotch candy from the United States of America.

I brought calendars of San Francisco, where I'm from and also San Francisco placemats, since placemats seem to be commonly used in Europe. Easy to pack, not very expensive and you bring a little piece of where you're from.