Luke tells a slightly different story of the Sermon on the Mount than Matthew's better-known version. This should not surprise or disturb us. Christians and skeptics are sometimes put off balance by discrepancies between the different gospel accounts. An Enlightenment based standard of exactitude combined with a defensiveness or skepticism that is constantly on the lookout for reasons not to believe creates this kind of problem for us.
In reality, there are good reasons why these accounts might differ. For one, Luke tells us that he used other source material, and he was no doubt familiar with Matthew's construction of the "Sermon on the Mount" (Matthew 5-7). But Matthew was writing for a largely Jewish audience, and Luke is writing as a Gentile for a Gentile audience, specifically a Roman official. That will shape the narrative. Another factor that is usually overlooked is that Jesus would likely have told the same stories and preached the same messages over and over again -- and it's entirely possible that Matthew's narrative reflects one of these occasions when Jesus was preaching on a hillside, and Luke has interviewed other sources who remember a time when Jesus preached a very similar message but on a "level place" (Luke 6:17). There is no necessary discrepancy. A third factor, of course, has to do with the nature of human memory and how different people experiencing a momentous event (a car accident, for example) will vary significantly in their recounting of the details, even though the main facts are the same. (This is why the varying accounts of the resurrection in the four gospels, for example, reinforce rather than contradict one another.) In short, we should not be disturbed by the variance between Matthew's and Luke's accounts of Jesus' teaching.
It is the content of the teaching that is truly remarkable. Here again, we see Jesus providing new wine that must be packaged in new wineskins. Jesus takes our preconceived notions of God and turns them on their heads. Where we imagine God as a righteous, perhaps vindictive, judge, Jesus says his Father is in fact merciful, pouring out blessing on the ungrateful and the evil. (BTW this is the same message, by implication, that alienated his townsfolk in Nazareth in Luke 4.)
Jesus lifts up -- declares "blessed" -- those the world is ready to discard, and castigates those the world envies. He expands the target of love immeasurably, insisting that love should include enemies, those who curse and abuse us. Reflect the nature of God, Jesus says, and stop creating God in your own image.
How are we to know such a God? How can we have an accurate picture of who this enemy-loving God is? This points us to the greatest mistake that is made with Jesus' teaching. You can't have Jesus' teaching without him being at the center of it. Jesus is the king of the kingdom he is proclaiming, and you can't have just comforting platitudes based on Jesus' message. Instead, this is a radical recasting of human existence under the uncompromising lordship of Jesus himself. In essence, Jesus says: I am coming as king. Here is what my kingdom is like.
At that point, as C.S. Lewis and others have pointed out, it is not workable to say Jesus is simply a great teacher. He is Lord, and these are the principles by which one lives in his kingdom -- or he is a self-absorbed lunatic who was rightly killed as a public menace. If he is a great teacher, the heart of his teaching is his own identity.
Christians have struggled mightily with these teachings over the centuries. Some have said this is an idealized system that is not attainable by human effort. Others have said that only a few elite Christians -- martyrs, or monks, or saints -- will be able to reach these lofty goals. Perhaps the reason these statements -- "Blessed are you who are poor, for yours is the kingdom of God" -- seem so far beyond us is because they reflect the values of God himself. The best explication of these teachings that I know is Dietrich Bonhoeffer's careful analysis in The Cost of Discipleship. Bonhoeffer lays out how these kingdom principles are serious guidelines that set a standard not only for the individual believer, but also for the Jesus-focused community. They are kingdom principles, and our churches will reflect them to the extent that we are kingdom communities.