John 3:3-8 is an important scripture to most Christians and to LDS Christians in particular. Most of my protestant friends have suggested that the scripture speaks of spiritual rebirth through the act of being born again by the Holy Spirit. LDS have generally interpreted the scripture to refer to the ordinances of baptism in water and receiving the gift of the Holy Ghost by the laying of hands. This interpretation was common also among the early Christians (see Born again translation).
In either interpretation, this passage of scripture is fairly important. Protestantism teaches that the act of being "born again" is the essence of Christianity itself. It is the one act for which nearly all sermons are preached and it is the very center of Protestant teaching.
On the other hand, LDS teaching generally assumes (except the teaching done by the missionaries) one has been born again already through baptism and the gift of the Holy Ghost. It is not the center of Christian teaching for LDS, but rather only the beginning of Christian teaching--the gate, the entrance into the kingdom, the first principles and ordinances of the gospel, the milk. There is much more to do and to understand after one is born again in LDS thought.
After Jesus finished telling Nicodemus that a man must be born of water and the Spirit, Nicodemus was confused. Jesus in response replied, "If I have told you earthly things, and ye believe not, how shall ye believe, if I tell you of heavenly things" (John 3:12)?
Does it seem odd to anyone else that Jesus would call being "born again" (the Greek actually means "born from above") "earthly things"? What are these "heavenly things" that he isn't talking about? From the Protestant perspective that being born again is the very heart and soul of Christianity, it is strange indeed. But from an LDS perspective that being born again is only the beginning, elementary steps of salvation, it makes perfect sense.
Interestingly, Ignatius wrote an epistle to his Christian friends in which he said,
"I am able to write to you of heavenly things, but I fear lest I should do you an injury. Know me from myself. For I am cautious lest ye should not be able to receive [such knowledge], and should be perplexed. For even I, not because I am in bonds, and am able to know heavenly things, and the places of angels, and the stations of the powers that are seen and that are not seen, am on this account a disciple; for I am far short of the perfection which is worthy of God" (Ignatius, Romans 9, in Ante-Nicene Fathers 1:104, brackets in original.).
So, these "heavenly things" are things reserved for some of those who have already become Christians, who have already been born again, who have already received "the earthly things."
I think there is a clearly parallel to LDS thought concerning entrance into the church through repentance, baptism, and the Holy Ghost and then the later temple ordinances. The first are "the earthly things" while the temple ordinances are the "heavenly things."