I went searching for the best way to remove these stains, I'll start by telling you that more bleach isn't it. Bleach only makes the pit stains more yellow.
Rit white-wash laundry treatment works, but at $2.49 per treatment its kind of expensive.
I tried ammonia+dishwashing soap, which was recommended by the University of Illinois Extension Stain Solutions Department. Results -- Failed.
I came across a product called Raise Yellow Stain Remover. To use it, you simply lay your shirt on a towel and apply a bit Raise on the armpit stains, brush the stain a bit, and then let sit for 15-20 minutes. Then drop the shirt in the wash as usual. This actually worked, but at $12.50 plus shipping, for a 12 oz bottle it was kind of pricey.
The best solution I found was OxiClean, the miracle cleaner. All you do is fill up a sink with warm water and mix in one scoop of OxiClean. Place the your blighted shirt in the sink, making sure the yellow stains are completely submerged. Let soak for an hour; tougher stains you might want to let sit overnight. Then launder as usual.
A 3 lb tub of OxiClean costs about $11.99 so this was by far the most effective and cost effective way to solve this nagging problem.
How to prevent stains in the first place
An ounce of prevention is worth of a pound of yellow arm pits. To keep the pesky stains from discoloring your nice white dress shirts you can wear an undershirt. But this does nothing to protect your t-shirts from being yellowed. To prevent this all together, stop using aluminum-based antiperspirants or for added dryness protection use your regular deodorant and Gold Bond powder. The Gold Bond does a good job of stopping wetness without using stain causing aluminum salts.
Another thought is to keep your arm pit hair trimmed to reduce wetness. Finally, you can wash your shirt immediately after wearing it. This prevents the stains from setting in.