Let's see what you got.
I'm trying to get an effect kind of like this, but without obscuring the background.
Why not obscure the bg?
By applying motion blur effects it'll definitely convey speed, and the player needs to focus on the foreground characters anyway.
What are you losing by obscuring the bg?
Like you mention, long dark think lines moving by at fast speeds in exactly the direction being traveled is one cartoonish technique, but I can't see the lines being interpreted as rain (or any other literal, physically existent thing) if animated right.
How about camera shake? I've often seen video reinforce a sense of fast speed by making the camera shake back and forth, as if the air is so turbulent it's rocking the camera.