We put down a jute rug in our dining room when we moved in and never thought about it again.

Now we noticed residue in a grid under the rug. Any ideas on what to use to clean it? I tried dry and damp scrubbing with a scotch pad, but that isn't doable. They're scraped red oak hardwood finished with what I think was a penetrating stain. Don't recall if Varathane or Minwax, but it was a blend of Jacobean and Ebony if that matters at all.

Any cleaning products that won't discolor the wood, or am I hosed?

Try a magic eraser in a less visible spot to see if it will discolor or damage the finish first. If it doesn't, try it plus hot water.

And a magic eraser is just a Melamine sponge that you can buy in bulk on Amazon. Dont buy the overpriced name brand.

Second is maby a hardwood safe floor cleaner and hot water. I use this diluted in my robot mop/vacuum for weekly cleaning. Search around to see what fits your floor best.
https://a.co/d/fDW04Ht