My approach is simple: get the kids to understand it and enjoy mythology. A lot of my students really struggle with anything in the unreality realm, so for some of them all this weird stuff and the weird names are just a bit much. First approach, get rid of the poetry.
Now I know, some of you might be saying, "But it's an epic, it is supposed to be poetry!" and you're right. However, since my main goal is to teach reading comprehension for the state test and I know that chances are slim that they'll put an epic poem on here, I'm willing to let the poetry thing slid. After all, once you've translated it from the ancient Greek, you've really lost the whole poetic feel to it. All you have left now is some translator's poetic feel, not Homer's.
Despite that, I expose them to the poetry int he invocation and the sirens. Otherwise we talk about it, read story excerpts from different books, read it in play form, and, of course, watch The Simpsons "Tales from the Public Domain" episode (just the first ten minutes) for their take on Odysseus's travels. If you are a teacher and are tired of the text book translation, scoot on over to my class page and you'll see all of the handouts that we used to read this beast. Oh, and you have to begin the unit by watching Star Wars: A New Hope. There is just no other way to teach it.
Feel free to leave a comment for suggestions on how to improve it! I was frantically creating it as we went along, trying to keep up with the class.