Actually, LDS believe that he did return. In our scripture called Doctrine and Covenants, 110th section, verses 13 through 16 read, "After this vision had closed, another great and glorious vision burst upon us; for Elijah the prophet, who was taken to heaven without tasting death, stood befure us, and said: Behold, the time has fully come, which was spoken of by the mouth of Malachi -- testifying that he [Elijah] should be sent, before the great and dreadful day of the Lord come -- To turn the hearts of the fathers to the children, and the children to the fathers, lest the whoe earth be smitten with a curse -- Therefore, the keys of this dispensation are committed into your hands; and by this ye may know that the great and dreadful day of the Lord is near, even at the doors."
This event went mostly unmarked by the rest of the world, but it was of tremendous importance in the grand scheme of things. Elijah did return and gave to the prophet Joseph Smith the keys and authority of beginning the last dispensation before the Second Coming.
Any thoughts on this from others?