I know it's kind of late for me to chime in, but oh well.
My brother and I don't wear any sort of deodorant. I'm a girl, so it's no big deal since I shave my pits. My brother wears short-sleeved undershirt underneath his t-shirt, he says that absorbs the moisture without having a wet t-shirt (or whatever kind of outer shirt). It works for him, even when he was living on the equator and now in the 100F summer heat.
The only time I had armpit odor was when I tried using deodorant for a while. I think those deodorants cause the odor most of the time.
I know it varies from person to person, you got some good suggestions, I guess one will work for you. If that doesn't work, you could go read some of the posts in the Body Odor forum, I'll bet they have some extra powerful ideas over there.