Used to work in a restaurant and what we used to do was to wash the table with hot water and washing up liquid, dry it thoroughly with a tea towel - loads more stains will show up.
Then leave for a few minutes and polish - worked on those surfaces although I can't remember what the surface was.
D
Had the same problem - I'm not an expert, but I think the rings are trapped moisture.
I used the steam from an iron to 'open up' the wood and a dry cloth to remove the moisture/white ring. Worked a treat and just polished the table when it had cooled.