A baking soda-water paste works well for me removing water rings from my unfinished wood table.
If the table is unfinished that means the water has got into the grain of the wood. You will need to bleach the wood surface to hopefully lighten the color of the stain. Otherwise you will need to do exactly what you do not want to do. And that is sand until it is removed.
Steve suggested baking soda and water paste, Works great on a finished top, but if there is nothing sealing the wood surface it will be a hard thing to fix without aggressively sanding and then finishing the surface.
It's iron that makes the black marks. Try a rust remover like Wink to get rid of them.
Worked when I had a table like that.
I know on a finished furniture piece when you get a ring you use mayonnaise. When the piece is unfinished the ring has penetrated the wood so either bleach or sanding is the only way I think will take it out. Usually unfinished furniture stores know how to fix your problem. Call and ask them. They may have a product you can purchase.