Actually those sound like exaggerations or even false accusations from fundamentalists. Catholics do read from the Bible, in fact it is widely accepted that the canonic gospels as we know them today are a product of the early Catholic Church.
Although I am currently a non-practicing atheist, I was raised a Catholic. I can tell for a fact that the priests tried their very best, stopping short of bribery, to get us to read or at least *skim through* the bible.
I remember that during mass, in addition to the sermon by the priest, there were usually public readings from the bible (mostly the new testament), and it was relatively honorific to be chosen to read before the congregation.
Last time I attended mass was during my wedding, twelve years ago. The priest quoted some verses from the Song of Songs (cliché) and read a beautiful passage from the new testament.
If Catholics have started asking their flock not to read the bible, it must have happened very recently.
I do know that Lutherans follow the bible as well. I know relatively little about the Amish and Orthodox religions, however I don't see why should the assertion be true about those two religions, when is demonstrably false about the other two.