We use tile nippers with the cutting edge ground back a little to keep them from cutting the head off the nail. You can get fairly long handled ones, and the nice round end gives you good leverage without really damaging the wood. It kind of "rolls" the nails out. If you can't get at the nail head at all, a "cat's paw" puller works, but will leave a pretty good hole.

As a last resort we use a 1/4" drill bit and drill right along the side of the nail on one or two sides. This tends to break the fiber bond and allow some movement of the spike.

Good luck, and remember your nail bucket!