Neither LDS chapels nor temples have crosses on their steeples, nor will you find a cross or crucifix inside. Why? The most commonly given reasons run along the lines given by Bruce R. McConkie, one of the LDS Church's apostles, just after the middle of the last century:

The sectarian world falsely supposes that the climax of his (Christ's) torture and suffering was on the cross . . . a view which they keep ever before them by the constant use of the cross as a religious symbol. The fact is that intense and severe as the suffering was on the cross, yet the great pains were endured in the Garden of Gethsemane. (Bruce R. McConkie, Mormon Doctrine 555)

McConkie bases his remark on what Jesus says in the LDS scripture, the Doctrine and Covenants. Speaking of his atoning suffering, he says: 

Which suffering caused myself, even God, the greatest of all, to tremble because of pain, and to bleed at every pore, and to suffer both body and spirit—and would that I might not drink the bitter cup, and shrink—nevertheless, glory be to the Father, and I partook and finished my preparations unto the children of men. (D&C 19:18-19)

That is the warrant for our belief that the suffering of the atonement took place on Gethsemane as well as on the cross. In 1975, Gordon B. Hinckley, then also an apostle, gave a different explanation. He said:

I do not wish to give offense to any of my Christian brethren who use the cross on the steeples of their cathedrals and at the altars of their chapels, who wear it on their vestments, and imprint it on their books and other literature. But for us, the cross is the symbol of the dying Christ, while our message is a declaration of the living Christ. (Gordon B. Hinckley, LDS General Conference, April 1975)

Since we understand Christ's atoning suffering to have begun and, perhaps, to have reached its climax in the Garden of Gethsemane, and since we want to emphasize Christ's resurrection rather than his death, we don't use the cross as a religious symbol. Those encompass most of the reasons that Mormons will give if asked about the absence of crosses (though we do use paintings of the crucifixion in our chapels as well as paintings of Christ suffering in Gethsemane).

But it appears that Mormons used the cross in a variety of ways, though not often as architectural ornament, until the early 20th century. Then, influenced by the anti-Catholicism of Protestant America and by our own home-grown anti-Catholicism, Mormons gradually began to feel less and less comfortable with the cross as a symbol. That feeling appears to have become institutionalized by the late 1950s. (A good summary of the history of Mormon use of the cross can be found here.)