Some time back, Kevin Barney ventured forth the opinion that the Church’s approach to atonement is a mix of many theories, especially pointing to how the hymns of the Church can be mapped to what Barney refers to as the four key theories of atonement. 
I believe Barney is correct in his assertion that the Church adopts a variety of explanations for how the atonement operates, but I’d like to further note that this is really the only option that a church attempting to holistically follow the Bible can make. The Bible itself does not have a central argument for explicitly how the atonement occurs, but rather has several contradictory metaphors.
In order to illustrate this point, I would first like to openly suggest that everything we know from the Bible regarding the atonement is primarily derived from the writings of Paul. Paul is the one who connects the dots between Old Testament forms of ritual and law and Christ. New Testament teachings regarding the atonement outside of Paul are typically seen either as derived from Paul (Hebrews, Letters of John) or as later additions to early texts (as in the last supper references). For those who say these points are arguable, that’s totally fair, but I think focusing on Paul can still get my point across.
Paul uses a variety of metaphors, drawn from his Jewish culture and background to explain the atonement. He uses sacrificial sprinkling , the sin-bearing scapegoat, a paschal lamb , heroic martyrdom , royal adoption , slave redemption  and conquering victor  just to name a few examples, and he often mixes and conflates metaphors.  As one author put it, he uses these concepts not to create a singular doctrine, but instead to create a “multiplicity of ideas” that “influence one another…but also contradict one another”. 
So while it is totally fair to say we do not have a singular atonement doctrine, it is only because there is no definitive doctrine to be had from God at this time. “I sense in a measure,” President Hinckley said, “the meaning of His atonement. I cannot comprehend it all. It is so vast in its reach, and yet so intimate in its affect that it defies comprehension.”  That is not to say that studying the atonement does not bear fruit , but that we have to expect that using our scriptures as our only prooftext in discussing the atonement is going to lead us to a veritable plethora of side roads which may come with as many issues as “the parable of the bicycle”. There are many, many ways to understand the atonement, and it does not hurt us to be open to the variety.
- I am adopting the term metaphor here based on Stephen Finlan’s usage of it- see “Options on atonement in Christian thought” by Stephen Finlan, Liturgical Press, 2007, partially available online here
- ritual purification, not penal substitution, like in Romans 3:25 NET where Christ is compared to the Mercy Seat [hilastrion] where the blood is sprinkled and also later to the animal whose blood is sprinkled. See also I Cor 5:7, I Cor 11:25
- the expulsion of sin via a cultic rite, not a sacrifice, but actually quite the opposite, like in Gal 3:13, II Cor 5:21, Rom 6:6, Rom 7:4, Rom 8:3
- protection from the wrath of God via the painting of blood on the door posts, not a temple sacrifice; see 1 Cor. 5:7-8
- a link to Maccabees [see IV Macc. 17:21-22] and messianism in general (not to mention Greeks like Socrates), where the deaths of some brought about salvation [liberty] for Israel. An example is Romans 5:7, also Rom 6:1-11; 8:10, Gal 2:20-21, I Thess 5:9-10
- signifying a change of status like in Romans 8:12-25
- also a change of status, but here involving a financial transaction or ransom like in I Cor 6:20, 7:23
- Victory over death and temptation, like in 1 Conrinthians 15:20-28
-“ Problems with atonement: the origins of, and controversy about, the atonement doctrine” by Stephen finlan, Liturgical Press 2005 available partially online here
- “ Jesus and the Doctrine of the Atonement: Biblical Notes on a Controversial Topic” Author: C. J. Den Heyer, Trinity Press, 1998
- “This was the most transcendent act that has ever taken place, yet it is the most difficult to understand. My reason for wanting to learn all I can about the Atonement is partly selfish: Our salvation depends on believing in and accepting the Atonement. Such acceptance requires a continual effort to understand it more fully.” James E. Faust, “The Atonement: Our Greatest Hope,” Ensign, Nov 2001, 18