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Yeah, @stenus, it's in Hareskoven next to the Furesø lake. There is a 25ish km marked mtb trail, a bunch of designated campsites and a few shelters. I've downloaded http://www.shelterapp.dk, which gives a great view of where you can camp. It covers all of Denmark, I think. On Singletrack.dk you can find mtb trails.
Rules in Danish public forests are: No camping, unless you're in a designated spot and no riding outside trails/paths.
Can you specify where this is? I'm moving to Copenhagen in September and will be looking to get out for some weekend adventures.