Mix three parts baking soda and one part water in a bowl. Stir until a thick paste forms.
Scrub the ugly sweat stains with the paste and an old toothbrush. Let it sit for 20 minutes. Yes, baking soda has other uses besides cooking.
Rinse the paste off with cold water, and then launder the clothing in cold water with an oxygenated laundry detergent. Hang the clothing outside in the sun to dry, if possible. If not, let it air-dry indoors. Don't dry the clothes in the dryer until you are sure the sweat stains are completely gone. If you do, you're going to have a harder time getting rid of those nasty stains.
Blot any stubborn, still-not-budging stains with a mixture of equal parts cool water and ammonia. Let it sit for a few minutes, then wash the clothing in cold water with an oxygenated laundry detergent.
Drop two aspirins (yes, even if you don't have a headache) in 1/2 cup cool water for yet another method for removing sweat stains. Stir with a spoon until the pills dissolve, then pour the solution onto the sweat stains. Let it sit for two hours, drizzle clear liquid laundry detergent onto the stains and rub it in with your fingers. Launder the clothing as usual.