I've worn nothing but baking soda as deodorant for a full week. The verdict? No smell. I was just sweating in the garden this morning--90 degrees and humid today. No smell. Nothing. I check my armpits often, and I actually enjoy the smell--I don't know how to describe it, but the word "fresh" comes to mind. Baking soda does freshen things up, like your refrigerator.
I like this a lot and hope it holds up. Baking soda is a chemical, sure, but probably one of the safest. I don't have to read ingredients or gamble one toxin versus another against my health. Its cheap and effective. And the big corporations don't want you to know. Most people have trouble believing that it can be this easy. An expensive product with lots of advertising must be better!
The theory is simple: Your sweat does not smell. The smell comes when bacteria feast on your sweat. So all you have to do is change the pH by making your armpits more basic (baking soda) or more acidic (vinegar) and bacteria won't grow.
I have had baking soda on for periods longer than 24 hours and still no smell.
To be very clear, there may have been a small hint of a smell on the first day I tried this. I may not have applied it properly on the first day. The smell was not on me, but rather on the underarm of my shirt. So I rubbed my shirt lightly on my armpits to pick up some baking soda residue. The next time a I checked, I didn't smell anything (it was so faint I might not have even had a smell--occasionally I catch a whiff of someone's food or something in the air, and I think its me sweating, but then when I check again I realize its something else).
Another alternative is to apply vinegar through a spray bottle to your armpits. I figure if the baking soda fails, perhaps the bacteria found a way to tolerate it. So if that happens, I would alternate vinegar for a while to offset that. I hear that you don't smell like vinegar when you do this--the smell dissipates quickly.
The skin irritation I was experiencing the first few days went away. I am not sure if that is because I applied two coats that day, rubbed it in too firmly, or whether my body adapted to it. I should also note that my armpits were already mildly irritated from using the Trader Joe's deodorant, so maybe the baking soda just made an existing condition worse, rather than generating it.
I am firmly in the Vinegar and Baking Soda Club. They can do almost anything. Clean your floors with it. Put it in your food. Deodorize yourself. I hear that spraying vinegar on weeds is as effective as Round-up--and certainly not a harm to you or the environment at large.
For more information and links to sources, see recent posts (and comments) on baking soda and another that is a write-up about Trader Joe's deodorant and issues of toxicity of deodorants in general.
UPDATE: See this more recent post on concerns about baking soda and a switch to vinegar.