The short answer: Yes, but I don't recommend it.
The unfortunate reality is that jewelry can become a trap for bacteria. In the case of rings, it likes to make its way between the band and your skin. While this is often harmless for the ring, it may not be in your best interest to wear that ring while showering. I think I speak for all of us when I say we don't want to go to the bathroom and have residue left over on our hands.
In addition, pretty much every soap/ conditioner/ and beauty product is likely to leave a film over your jewelry. I've always noticed this the most when it comes gemstones of any kind. The residue usually leaves the gem looking cloudy. While this isn't going to damage your piece in the short term, many people agree that the appearance is undesirable. Unless you know how to clean your stainless jewelry, you might find it tiresome to have a profesional polish it this often.
When it comes to bracelets and necklaces we have a little more leeway. These aren't as much of a health risk. Most of the time all you have to worry about is any films that will be left behind on the piece of jewelry itself.
As for piercings, I'm not the best guy to ask. My experience with piercings resulted in every piercing not only not healing but also becoming infected and needing to be removed.
Will stainless steel jewelry rust?
What I've found both in practice and in the research literature is that sweat is far more likely to result in the corrosion of metals (this includes stainless steel) than the water coming out of your shower head. If you're interested you can read more on this here.
Generally, take your jewelry off when you don't need to be wearing it. For rings, take them off to wash hands and before going to the bathroom. For a more exhaustive list on what not to do with your ring, I recommend this article.