I have the same problem, and am looking for a different material non-skid rug pad for the future, will let you know what I find. You posted this some time ago, don't know if you found a solution, but for your future ref:
To clean, best product is Bestine Rubber Cement Thinner. You will find this in any art supply store ( at utrechtart.com it's $8.99 a pint, get at least a couple depending on how big an area).
Wear protective gloves, and apply with large cotton pad. I would flood an area of the floor and just begin wiping away in a buffing motion, may take a couple times with fresh cotton pad and cleaner - It will remove the rubber without damaging the floor finish.
This stuff is great in general to have on hand to remove any sticky residue from price stickers, labels, etc. I have even used it on fabrics....Have never had it destroy a surface (not like rubbing alcohol which will cloud plastics and doesn't do a good job anyhow)