The Church of Jesus Christ of Latter-Day Saints believes in the doctrine of grace. In fact, since it's part of the Bible, Latter-Day Saints believe in Eph 2:8-10, which says:
8 For by grace are ye saved through faith; and that not of yourselves: it is the gift of God:
9 Not of works, lest any man should boast.
10 For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them.
Usually, the LDS Church focuses more on the good works part, since their teaching is focused on living their faith practically, but they fully acknowledge their complete reliance on the atonement of Jesus Christ. Without it they would be lost forever. In fact, the Book of Mormon supports the doctrine of grace in many places. Here is 2 Nephi 2:4,6-8:
4 [...] And the way is prepared from the fall of man, and salvation is free.
6 Wherefore, redemption cometh in and through the Holy Messiah; for he is full of grace and truth.
7 Behold, he offereth himself a sacrifice for sin, to answer the ends of the law, unto all those who have a broken heart and a contrite spirit; and unto none else can the ends of the law be answered.
8 Wherefore, how great the importance to make these things known unto the inhabitants of the earth, that they may know that there is no flesh that can dwell in the presence of God, save it be through the merits, and mercy, and grace of the Holy Messiah, who layeth down his life according to the flesh, and taketh it again by the power of the Spirit, that he may bring to pass the resurrection of the dead, being the first that should rise.
However, Latter-Day Saints believe that there is no such thing as cheap grace. Faith in the sense of merely believing that Jesus Christ is the Son of God without a change of heart (that usually is manifested in good works) does not lead to salvation. Latter-Day Saints believe that the following things are necessary to be saved:
- faith in the Lord Jesus Christ,
- repentance (meaning a change of heart and turning towards God),
- making a covenant with God to take upon yourself the name of Jesus Christ through baptism,
- receiving the gift of the Holy Ghost through the laying on of hands (Acts 8:14-17; 19:6),
- keeping the baptismal covenant by pressing forward with a steadfastness in Christ and enduring to the end.
The need for baptism and receiving the gift of the Holy Ghost is often explained by John 3:5:
Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.
One might argue, that it is not really grace, if you have to do something for it. But continuing to live in sin after having received the truth is like making the Son of God the servant of sin. Speaking of grace, Paul wrote the following in Romans 6:1-4:
1 What shall we say then? Shall we continue in sin, that grace may abound?
2 God forbid. How shall we, that are dead to sin, live any longer therein?
3 Know ye not, that so many of us as were baptized into Jesus Christ were baptized into his death?
4 Therefore we are buried with him by baptism into death: that like as Christ was raised up from the dead by the glory of the Father, even so we also should walk in newness of life.
If you want to study the teaching and understanding of grace in the LDS Church better, then I can recommend this talk by Brad Wilcox as well as this article by Bruce C. Hafen.