There are lots of "new" products out there that will handle this problem effectively. Try some of the products designed for washing/bleaching nappies. These are now "colour safe" and shouldn't bleach your shirt at all. In Australia there's a product called NapiSan that does exactly this. It's just a powder that you add to the wash. Alternatively you can add a small amount of water to the powder and make a paste to rub onto the stain before washing.
But, despite the new products, sometimes the "old" tips are best. For perspiration stains try vinegar or, for very stubborn stains, try cream of tartar and some disolved aspirin. I'm not kidding! Once you use vinegar just wash normally and this will remove both the perspiration smell, but also that of the vinegar.
Check out the website
http://www.chemistry.co.nz/stain_frame.htm. It was created by a chemist and describes some great methods of removing hundreds of different kinds of stains easily.
Hope this helps.
Mr Mephisto