When you hear bad news (affecting the person talking to you) do you say "I'm sorry" or "I'm sorry to hear it."?

I used to just say, "I'm sorry," but too many people jumped in with, "It's not YOUR fault" (that Grandma died, that the dog ran away, etc.), so I switched to, "I'm sorry to hear it." I'm not entirely comfortable with this, though, because it seems to state that, I would be fine with the thing happening, if only the person would not bother me with the news. Of course, this is not what I wish to convey at all.