I've only ridden in Copenhagen and Aarhus, but it really was a great time. Aarhus has more hills than you'd expect in Denmark. Even if you're set on touring the whole country, you should set aside some time in Copenhagen just to get into the urban design and bike culture. The food these days is as fantastic as you hear.
One thing to keep in mind: the latitude of Copenhagen is 10 degrees north of Portland, Maine -- in the summer, the sun comes up really early. Bring an eyeshade! Also, the weather can turn cold and rainy, even in August. And the wind is real.