It depends on which theme you want to discuss. Probably the most obvious theme is the decline of the Old South. Many of Faulkner's stories and novels wrestle with the challenge the South faced as old traditions and values were replaced. This is revealed in the opening paragraphs as the narrator (probably a member of the community or a composite of people in the community) describes Emily's dilapidated old house which sat on "what had once been one of our most select streets" but is now an "eyesore among eyesores." The Grierson family had once been considered among the elite of the town, but the father's death, Miss Emily's bizarre reaction to the death, and her courtship with Homer Barron (a loathsome Yankee) each contributes to the demise of the Grierson family name.
The setting itself helps underscore this theme. As the story progresses, the town of Jefferson evolves into the modern era: roads are paved, numbers are attached to houses, and sidewalks are put in. However, Miss Emily's house, and her own attitudes for that matter (note the scene when she is presented a bill for her taxes), fail to change with the times. Furthermore, the old guard (Col. Sartoris) is replaced by "the rising generation." Note the scene where the younger men are left with the task of sprinkling lime about her house to get rid of the smell. They aren't as in tune with the sensibilities of dealing with an eccentric southern woman, as Judge Stevens schools the young generation, "Dammit Sir...will you accuse a lady to her face of smelling bad?" All of these examples serve to illustrate Faulkner's take of the decline of the old South and the slow, gradual birth of the new South.
Another possible theme is the basic human desire for love and companionship. It is made clear that Emily's father ruled her young life. "None of the young men were quite good enough for Miss Emily and such." After losing her father, Miss Emily (already past thirty) is faced with a life of spinsterhood. Only after she meets Homer Barron does she reveal a glimpse of happiness. However, when she realizes that he is "not a marrying man," Miss Emily decides to "cling to that which had robbed her" so she poisons Barron and places his corpse in her bed. Her father already left her; this one will not get away!
(....which brings up quite a debate about Homer Barron in my classroom. The text says, "he liked men, and it was known that he drank with the younger men in the Elks club--that he was not a marrying man." My students say he is either gay or he just doesn't want to be tied down. My own thoughts are that including a homosexual character in a story written in the 1930's when being gay wasn't quite as easy as it is today was probably not something Faulkner entertained.)
Perverted as it may be, Miss Emily's desperate attempt to hold on to her only love reveals the lengths some people will go to in order to have love in their lives.