Others can chime in, but it's hit or miss for me. I do have one white long sleeve knit from Banana Republic that has never gotten armpit stains, and I've had it for years. It has, however, started to loose it's luster overall... a bit of pilling... so I only use it as a layering piece now.
GAP tees - I do love them because they fit me very well (now that they have petites, even better!) But the white shirts never last more than one season. You don't even see the pit stains until the next year when you bring them out.
I have tried expensive white tees (blanking on the brand now, but it's sold on bluefly.com) and they also got pit stains after one season (appearing after being in storage).
The colored tees do better, but they do eventually loose their shape.... whether worn and washed, or just kept in storage. Black tees don't stay black either.
I do keep a stable of GAP tees for layering or just lounging around the house or for boating in the summer and I just buy on sale, and then replace them every year.
Your linen shirts should work very well with the method I described!! I have had the best luck with linen. Something about hanging them to dry in the sun brings back their whiteness beautifully.