We are often taken to task for our use of several Bible passages that appear to support LDS doctrines in this area. The three most relevant passages are 1 Peter 3:18-20; 1 Peter 4:6, and 1 Corinthians 15:29.
1 Peter 4:6:
For for this cause was the gospel preached also to them that are dead, that they might be judged according to men in the flesh, but live according to God in the spirit.
1 Peter 3:18-20:
18 For Christ also hath once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit:
19 By which also he went and preached unto the spirits in prison;
20 Which sometime were disobedient, when once the longsuffering of God waited in the days of Noah, while the ark was a preparing, wherein few, that is, eight souls were saved by water.
1 Corinthians 15:29:
Else what shall they do which are baptized for the dead, if the dead rise not at all? why are they then baptized for the dead?
If you're interested in exploring some of the arguments made against the LDS use of these verses, especially the first two passages, a good resource is a review by John Tvedtnes, "The Dead Shall Hear the Voice." He gets into some common anti-Mormon nitpicking and shows that the LDS view is consistent with the Bible and early Christian thinking as well.
For fascinating details from early Christianity regarding the spirit world, preaching of the Gospel to the dead, and baptism for the dead, you may enjoy Chapter 4 of Barry Bickmore's outstanding book, Restoring the Ancient Church. Highly recommended!