[Roosh]

http://www.dba.dk (also great for getting a used bike). Use Google Translate to navigate the site

Couchsurfing rental groups for Copenhagen:
http://www.couchsurfing.org/group.html?gid=2996
http://www.couchsurfing.org/group.html?gid=11434
http://www.couchsurfing.org/group.html?gid=22276

http://copenhagen.en.craigslist.dk/hhh/