Apparently, it's not the anti-perspirant, so much as what happens to your anti-perspirant in the dryer. Your same question was answered by a dry-cleaner in a GQ "style questions" type column a couple years ago. They said that the heat of the dryer leads to a reaction with the alumnimum in the anti-perspirant that causes the staining.
I tested out their reasoning on two white shirts, drying one in the dryer and hanging the other on a line for a coule weeks, and the line dried one didn't get anything resembling a stain.
with that said, I've been using arm and hammer antiperspirant lately, and haven't noticed much staining at all (and i'm drying shirts in the dryer).