How do I remove white rings from my table?

My mom used to use this thing called an almond stick. I googled it and found it here Cool Tool: Tibet Almond Stick

She had some really good wood furniture that my brother and sisters and I put water rings on and rubbing this over them makes them temporarily disappear. Use it after dusting and it should diminish the rings.

Other than refinishing, I don't think anything will fix the problem.