I have not made a thorough study of world religions (I am a member of the Church of Jesus Christ of Latter-Day Saints – a “Mormon”) but I have done some light reading.
It is my understanding that many christian (protestant) religions do in fact teach that God stopped speaking to the world shortly after Jesus Christ walked the earth. Various reasons are given for the silence, from “He’s said everything He has to say” to “We’re too wicked for Him to talk to us”.
The LDS teaching is that God speaks through a “priesthood” power, and that shortly after the time of Christ the anti-christian sentiment resulted in the death of everyone who held that priesthood. The priesthood can only be given by someone already holding it, so once it’s gone, it’s gone.
The world existed in that state for several hundred years, until Joseph Smith was given the priesthood by a resurrected John the Baptist and by Peter, James, and John of New Testament fame. At that point, God could again speak to us, and the Mormon canon of scripture includes numerous revelations to Joseph Smith and subsequent church leaders.
(This doesn’t answer the question of why He waited nearly two millennia to restore the priesthood. The official answer is that the world wasn’t ready – that a restoration at any earlier time would not have been permanent. This may be supported by the hostility that early Mormons were subjected to. If the restored church was persecuted so heavily in the land of Freedom, it’s perfectly reasonable to suppose that it would have been snuffed out of existence in a less tolerant environment).
The Book of Mormon is actually an inspired translation of the writings of prophets who lived in the ancient Americas. The compilation of modern revelation is called The Doctrine and Covenants. Mormons also believe that all of the teachings of our prophets are scripture when they speak “in the name of the Lord”, even if those teachings aren’t part of the official canon of scripture).