This passage occurs in the Revised Common Lectionary, Year B (Most recently Oct 2, 2015).
Summary: This is a very difficult passage, causing shame for many and perhaps even smugness for some. Many commentaries have been written about it. I'd like to focus on a few Greek words, especially some "απο" words, that might provide a framework for considering divorce and preaching about it. Again, very tough because everyone has such different baggage on this topic.
Side comment: Another helping tool for looking at these passages is to compare the words of Jesus in Matthew 19 and Luke 16 (if not 1 Cor 7).
πειραζοντες ("test or tempt", from πειραζω, 10:2) We see this come up often in the Gospels, where the Pharisees (or some other group) are trying to test Jesus. This case is a bit different. John the Baptist was imprisoned because he spoke out against the marriage practices of Herod. The Pharisees questions are intended to have Jesus imprisoned, if not killed. Our society has great culture wars going on now about marriage; perhaps each one of us will face persecution for our views. Lastly, if we wonder why Jesus is so harsh in his words, it is because the Pharisees are inviting him to his death.
αποστατιον ("divorce", 10:4) This word "explodes" off the page if you look at it in the Bible or in the Greek language. First, in Greek, this word meant leave one's station ("απο" means away; just read the letters in the word: a-p-o-s-t-a-t-i-o-n!). It meant a military defection from your captain, the one ahead of you in rank. Moses gave permission to write a certificate of defection!! What if we started calling divorce defection?? Ouch.
Jesus actually changes the law here. If you look up the word, you are taken to Deuteronomy 24:1, where a man is given permission to kick out his wife if she doesn't please him. Jesus today is calling men to a greater level of faithfulness than previous generations ever did; men cannot simply leave their wives because they don't please them! Jesus also even admits the reality that women might leave their husbands on their own accord, something unthinkable. In this way, Jesus alters the law (a radical concept), even enfranchising women, but finally asks for greater commitment. (Note however, that even though the Bible's teaching divorce shifts over time, the teaching on marriage remains the same).
απολυω ("free"; 10:4) This word can mean "release" or even finally "divorce" but it is worth looking simply at what the word means: to set free. As a pastor, I have seen this, where divorce is a freeing of someone from an abusive and unfaithful relationship plagued by addiction and anger.
So here is the million dollar question: When is the divorce "αποστατιον", namely, a defection? And when is it a απολυω, a freeing?
σκηλροκαρδια ("hardness of hearts"; 10:5) The word here contains the root "σκηλρος" which means hardness -- an awful disease is "multiple sclerosis", the hardening of certain body parts until finally the person cannot move. In a downward spiraling relationship, there is a hardening of the heart, until finally the person cannot love. As Christians, we believe that God creates new hearts (Psalm 51); however, Jesus admits (see also Matt 19) that certain conditions, like adultery, create such a hard heart, that the two are permitted not to be yolked any more. I would add abuse and addiction, both forms of adultery, you could argue, to this list of permissible divorces.
αρχη ("beginning"; 10:6) Jesus affirms that marriage is a long-long, committed relationship between a man and a woman, grounded in creation and the particular creation accounts we have in the Bible. This means that marriage has a few purposes: to offer companionship, to create new families and bring a couple into full intimacy, even union. I think one could further argue that marriage is a tool of God's sanctification in us, in that we discover our sinfulness very clearly, need forgiveness and become of great use to God through the love given to us by our spouse. Jesus returns the focus to God's goodness and intentions for marriage.
My haunch: The Christian church needs to spend a great deal of time and teaching on what marriage truly is and what it is for.