On a friend's advice I took a swing after work and checked out Kokkedal (Skovengen and Byengen neighborhoods) and then down through Horsholm and Rungsted and along Strandvel all the way through Vedbaek, Skodsborg, Klampenborg, Charlottenlund, Hellerup into Osterbro and CPH. I deliberately got lost just to see where I'd end up but I found the right neighborhoods anyway, it's quite easy :).
The Kokkedal neighborhoods were nice; Two-story row-apartments with some green space at back and pedestrian path separating blocks. Also there were some woods and field nearby that looked good for walking. It looked quite far away from transit though.
Horsholm looked very suburban. Rungsted was nice, very New England with older buildings and lots of deciduous trees. The drive down Strandvej back to CPH was also very beautiful with a big, red, full moon over the Oresund and Sweden on my right.
I can see why so many people have recommended Hellerup and environs to me - a really nice city neighborhood, that, but expensive.
Tomorrow I'll try Birkerod and points west and see what they're like.