Okay, I'm late to the party, but I'll throw my 2c worth in too.
My wife was out of town for a few months (2 weeks would be enough), so I used that time to adjust my body to no shampoo and no soap. I do wash my hands with soap, but that's all.
You'll feel greasy for a few days, but it passes. It is supposedly better for your body, and you skin adjusts to not having the protective oils stripped every time you shower. So, I shower twice a day most days (once on days without a workout).
I'm clean and don't stink. My wife comes home tomorrow. I don't think she'll notice a difference.
I haven't used antiperspirant in over a decade, but I still used deodorant until recently. No more.