How is Homer's The Odyssey relevant to American culture today?
While it is true that Homer's epic, The Odyssey, is set in an unfamiliar time and place and contains Greek gods and goddesses which are no longer relevant to modern life, there are many elements of the story which are quite applicable to modern American life.
One relatable element is loyalty. This story would not even have happened if Odysseus had not been loyal to his friend Menelaus and gone with him to rescue Helen. On a personal level, Odysseus has many people who remain loyal to him while he is gone (Eumaeus and Eurycleia, for example), and they assist him when he returns to reclaim his family and holdings. Despite his lapses in loyalty with some of the women and goddesses he meets on his journey, Odysseus is loyal to Penelope, and she is certainly loyal to him. Americans, too, appreciate and admire loyalty in their friends and families.
A second familiar element of this story is excessive pride. While people in the modern world do not regularly converse with gods (or make them angry), most of us have suffered the consequences of being too prideful, like Odysseus. (Those too proud to ask for directions often remain lost, for example.)
Another aspect of the story which modern Americans share with the characters in The Odyssey is temptation. So many times Odysseus' journey is prolonged because he or his men succumb to temptation. Each time, the consequence is something different; however, there is always a cost for giving in to temptation. (When the men eat the Lotus fruit, for example, they do not remember anything about their homes or their families, which of course prolongs their journey.)
So many other elements of this story are different but recognizable in today's culture. Once you begin to look beyond the specific settings and characters of The Odyssey, you will discover that this is just a story about human beings, their strengths, and their flaws.