PM'ed you.
The month of May is a wonderful time to be visiting Denmark as the weather would be really pleasant given the transition from Spring to Summer. =) It's the time of year when all the Danes start coming out of their homes, onto the beaches, into the parks and organizing open-air bbqs, parties and festivals. Depending on which city you'll be visiting, there will be various festivals and events going on which you might be interested in catching.
Since I'm more acquainted with Copenhagen, I've a couple of places which you should definitely check out.
Off the top of my head, a must visit is Christiania, a free-town (if you do not know what this means, you'll know in time to come
) located in Christianhavn, 1 metro stop away from the city centre in Kongens Nytorv. Just go google up on it and you'll be greeted with a plethora of information on what the place (Christiania) is about. Christianhavn as a location in itself is characterized by canals and bridges and low rise apartments, much like Amsterdam. To be honest, Copenhagen as a city holds alot of charm, and while being a very lively and active night-life city, still possesses a homely feel to it.
Going in the month of May means that there is another huge attraction that is open in town. Tivoli, the flagship theme park in Scandinavia, will come out of winter hibernation and burst into life during the period that you're there. It really is a glorious place where there will be live concerts organized every week, and not forgetting that it is still a fun-packed theme park