Most of the particle types are only visible in hardware renders. You need to render a separate rain layer and composite.

There are a few types of software rendered particle types, whose names in the list end with (s/w). These will render fine along with everything else, but it can be tricky to get nice streaky looks. If you MUST do a software render (say, because you have no compositing software at all) you might be able to get what you need with instanced geometry or by applying motion blur to software-rendered particles.

-- Mark