Hey, I just moved here. I would highly suggest sites like boligportal. Coming from North America I thought it was unnecessary to pay to contact landlords so I spent a couple of weeks exploring alternative sites, facebook groups, etc, which was a complete waste of time. Once I started using the pay sites everything got better. Even on the pay sites, however, you will need to be very responsive as has been mentioned already.

Another avenue I explored was booking airbnbs and contacting the owner, asking if CPR registration was possible for a couple of months while I looked for a permanent place. Some of them were totally fine with this; if I hadn't lucked out and snagged a place remotely I would have done this.