I used to brew with my tap water that was treated with a water softener, but I got a nagging unpleasantly bitter aftertaste in all of my beers. It wasn't salty like you describe, but it wasn't ideal, either.
In my case I just started splitting up my brewing water. I'll use about half tap water and half distilled water in my brew to get a more balanced water profile.
There is some debate as to what effect water softener water has on the overall taste, but without a true water test showing what's in it, it's really a guessing game. I just took the conservative route and started introducing some distilled water into the mix and am happy with the results.