What's the first thing I do (after reminding children that water and wood don't mix?)--- head to Pinterest! It turns out there are a few household remedies for water rings on wood.
I had much success with the first one I tried-- Mayo!
Just spread some mayo on the water ring and rub it in and wipe away. I did this about three times, and each time the ring got lighter. After I walked away and came back about 1/2 hour later, the spot was completely gone!
I wasn't sure how well this would work on older water ring stains, but I tried it on a spot on our kitchen table and it worked there too!