This passage occurs in the RCL, Year C, most recently May 29, 2016.
Summary: Like so many passages in Luke, there is layer of meaning regarding faith, healing and the Word of God; there is also another layer of complex social dynamics. Luke presents cultural rules and norms that are both being obeyed and broken. How to preach this?
One possibility is to ignore the social dynamics and focus on faith and healing (ie, preach as if you were preaching from Matthew's Gospel, in which the story is simpler!)
1) Jesus heals, even through the prayers of others;
2) Faith in Jesus changes everything; outsiders can have faith too.
Another way is to portray Jesus action over-and-against the social reality of his day. The world then and today is a messy, complex and broken place. The world is one of haves and have-nots; of powerful people with agendas (...in those days a decree went out from Emperor...) In spite of all of this, Jesus compassion and power triumph!
λαος ("people", 7:1). The word means "the people", as in the commoners. Luke pays careful attention to the λαος (36 references; Matthew 14; Mark 3 and John 2). This word sets up quite a contrast to Jesus interactions the rest of the pericope, where he is dealing with the leaders, religiously and politically. This reminds us that while Jesus cares for the commoners, he also cares about the leaders too. Compared to him, we are all chumps ;-)
δουλος ("servant" or "slave", 7:2) Because American history is defined by our freedom from England and then the freedom of slaves, we tend to value "freedom" greatly. Furthermore, we look with disgust on the entire concept of slavery. While I do not defend slavery, it is worth pointing out that within Greco-Roman culture slavery meant something different than American antebellum plantation-style slavery. At the very least, not all slaves were abused and many were considered part of the house. The centurion will even call the slave his "παις" or child; he considers the slave "εντιμος" or honored; so honored in fact, he seeks out Jesus' healing. This is a reminder that economic and social boundaries both then and today are often complex. More generally, the whole scene is one that really puts the preacher in a tough spot -- it is clearly a different world, one that we cannot imagine. An occupying army general asks the local Jewish healer for a favor regarding his boy-slave and then is found, bizarrely, to have more faith than anyone.
διασωζω ("save" or "heal"; 7:3) The root word here is σωζω, or save. It has dia- as a prefix. This prefix can intensify a verb, like adding the adverb "thoroughly." The point is that Jesus' salvation includes earthly healings.
αξιος ("worthy"; 7:4; appears later as a verb in 7:7) A reminder of the honor-shame dynamics in this culture (of which I know little). I do feel comfortable making two points though. First, it seems questionable whether Jesus should have been doing this healing for a non-Jew, especially a member of the opposing army. In fact, one must wonder about the relationship between the Centurion and Jewish leaders; could then even speak to each other directly? This is a difficult point for us to address or even consider as Americans. Second, Jesus power is overturning the cultural expectations of everyone.
πιστις ("faith"; 7:9) A reminder that faith is not a belief in a set of abstract principles, but trust in the divinity of Christ and the salvation he brings.
Two small notes on verb construction that point toward something deeper:
παρακαλεω ("encourage", 7:4). This verb is in the imperfect suggesting repeated action. It is unclear why they needed to repeat the request -- perhaps because they felt it important, or because Jesus didn't want to do it. But something about their continued urging moves Jesus.
μη σκυλλου ("no longer be troubled", 7:6) A reminder about the negative present imperative: μη + present imperative means "no longer" ie, you were doing this, but stop and continue to stop this. (Often used in the construction "No longer be afraid" when angels begin speaking to humans.)
Lastly, two words that come into English related to health
υγιαινω -- "hygiene" (the υ has a rough breathing mark)
ιαομαι -- "iatry" like "psychiatry"