In the past, I would scour my kitchen sink and use scouring powder on the counters. There would still be stains on the counters, so I did what my mom always did: put some bleach on the counter, then wipe and rinse. I disliked using bleach, since it has such awful fumes and I felt I had to rinse and rinse and rinse to get it all off the counter after the stains had faded.
So, here is my new, non-toxic method for removing stains from my counters. This idea was from 200 Fabulous Frugal Uses for Baking Soda by Vicki Lansky, which is also where I found my wonderful homemade scouring powder recipe [0].

First, here is a small section of my kitchen counter, prior to being cleaned using this method. These are tea and cooking stains.

Step 1: Pour or squeeze lemon juice over the stains. Supposedly a fresh lemon works best, but I save a little money by using the lemon juice from concentrate, and still get great results!

Step 2: After the lemon juice has soaked on the stains for a few minutes, sprinkle baking soda over the lemon juice. Wait another minute or so, while it stops fizzing. :)

Step 3: Using an old dish cloth or rag, gently rub the baking soda and lemon juice into any remaining stains. Stains should easily wipe off. Rinse your rag and countertop.
Ready to try it? Let me know how it works for you! :)
Note: While this method most likely won't harm your counter top, please test in a small area before you do an entire counter, just to be safe. It works for me! [1] :)
Links:
[1] http://rocksinmydryer.typepad.com/shannon/2007/03/works_for_me_th.html