The basic term is showerhead in English, and pomme de douche in French.
[Deleted this part: I didn't know 'douche de tête' but there is an explanation below]
Rainshower alone just sounds like a brief period of rain (outside), although you can get rainshower-effect showerheads. If you specifically need to translate "rainshower(-effect) showerhead", a native French speaker might need to help rather than me
thank you @ph_l your translation "pomme (de douche) à jet pluie" seems good.
I have checked research numbers with google and:
pomme de douche pluie - 110 (every month) - abbreviation of your version
pomme de douche à jet pluie - <10 (every month)
I have seen french sites that use "Douche de tête" like: goo.gl/JaxgGJ