Mountain bike the Lincoln Woods Trail, do the 4000 footers and don't worry about the recognition.
Actually, there is a way to mountain bike in and out and STILL be an official 4000 footer in the eyes of the AMC.
As much as I am a knob-head (knobby tires that is), I have to agree with Dave. The phrase: "...that are not part of officially maintained trails..." pretty much locks it up. Lincoln Woods is an officially maintained trail. Therefore, while you can legally mountain bike in to the 3 mile mark, that would exclude any peak that you hike from counting on an official 4000 footer list, as defined by the AMC.
HOWEVER, there is a pretty good sized loophole regarding the East Branch Loophole. The 26th White Mountain Guide officially refers to this route as the East Branch Truck Road (FR 87). While cars are no longer allowed on this ROAD, it still still designated as such. It is not a maintained trail, but rather a maintained road. The White Mountain Guide never calls this a trail, but rather refers to it as a road on several occasions, going as far to say that it is maintained for vehicle use, but allows pedestrian traffic.
By using the AMC rules, I would say that it is perfectly OK to mountain bike on the East Branch Truck Road, stash your bike at the Wilderness bounds, ford the Pemi, and head on to Owl's Head. Of course, this is all probably more work than just hiking the 3 miles in on the Lincoln Woods Trail, but would it be as much fun? Of course, all this only matters if you want to apply for recognition on the official 4000 footer list - something that about half of the 4000 footers have never done, myself included.
Either way, enjoy!
Oh, and if you want a really fun mountain bike, try heading up to Greely Ponds from Livermore. You have to stop at the Scenic Area sign about .5 miles from the pond, but you can get most of the way in there. The ride down looks like great fun.