Thanks to you both for the hints. I tried some mineral spirits on the last layer of residue, and it worked well. Unfortunately, the pattern of the carpet pad has "burned" onto the floor. Like I mentioned, we will be refinishing the floors during our renovation next year. For now, an area rug will cover the floor.
The peel away project looks interesting. Do you know of a national store that might carry it (Home Depot, etc.) so I can get some this weekend? If not, I guess I'll call the company to find a distributor.