I did constant rain on my first short. We used a couple of different techniques.
For the big exterior daytime scenes where the shot was pretty wide we used a fire truck. (There's an organisation in the UK called Young Firefigthers and they're always happy to help out, for a few hours)
For the night sequences we used a watering can. So after wetting down the parts of the set that could be seen, we dropped the water from the can across the front of the lens (about a foot ahead of it) and then we cross lit the water. Basicaly we put a small pag light to one side of the camera at 90 degrees the lens and about four foot back.
For the very tight night exteriors, which were also hand held, walking backwads with the character with had a runner with one of those plant sprays to the right of the camera op, putting a fine mist of water onto the character as they walked.
All these techniques worked really well and we won a Royal Television Society Commendation for the film.
Like many things it's actually more about giving the audience the right audio and visual clues and their imagination fills in the rest.