I'm going to go all sci-fi & fantasy with it:
There's the sonic shower from star trek. Similarly vibe showers from Babylon 5. These work using vibration and sound to take dirt grime and sweat off the body.
The dust shower from Tank Girl. Likely chemicals which kill bacteria in the dust. But this actually has precedent in real-life. Clean sand or dust can be used to cleanse the body when no water is around for ritual purification in tayammum. Seems counter-intuitive but desert cultures have used sand to scrub the body and clean it in the past when water is at a premium.
There's also the oil bath from Wicked, because the Wicked Witch is allergic to water. In real life, oil baths are sometimes used because of medical conditions and on the face. Here's a link to an abstract on a scientific paper on that and the effects on skin. In it, basically they talk about skin staying more hydrated as a result, which might be great in a desert environment. And, while this is ancient rather than not, look into the Etruscan/Greek/Roman strigil. Even though this oil-based cleaning is meant to be used with water in mind for later, it's still an interesting cultural touchstone.
In the series The Girl from Tomorrow (an Australian TV show in the 1990s) I believe there was a light shower, that is a shower using light to kill everything and take away grime. This concept has come up now and again in sci-fi. Now, in real life there's something called Ultraviolet germicidal irradiation but that's mostly used for equipment, water, and air in medical facilities. Because, if you look at the wiki link, it's sort of detrimental to human life, what with the cancer-causing and eye damage.