Another way to introduce this play would be to ask your students to write in their journals about a time they felt trapped and what their dream of escape was. For example, one student might write that she feels trapped living at home with her parents and longs for the freedom to have an apartment of her own. A more powerful way to give meaning to this exercise would be to encourage the students to use concrete authentic detail, perhaps even a specific anecdote, in telling this story in order to make it a good exercise in personal narrative writing. Exploring these themes in a personal way before reading the play would provide a matrix for and link to understanding it.
Although I try not to recommend videos as a way to introduce novels (I usually find that once students know there is a video, all hope of reading goes out the window), this video with members of New York's Actor's Studio, including Shelley Winters, Sandy Dennis, and Geraldine Page, is very interesting. Perhaps you could show them just a clip of it to get them interested. It was filmed in 1965, directed by Paul Bogart, and released by Hen's Tooth Video in 1998. There is also a superb 1970 film with Laurence Olivier which might be easier to find as it is much more well known.
I'm a student in a Russian Literature course and our teacher introduced us to "Three Sisters" by first showing us the movie Uncle Vanya which has a similar context to that of the "Three Sisters" and we then discussed it and the context of being middle class outside of Moscow and how different culturally the two places were. We discussed the hopelessness of being stuck in a society as a woman where they could not do anything and then we moved onto the play with some really good historical and textual context. Finally, we ended by actually getting to see the play and it was one of the best experiences with literature I have ever had.
P.S.- Below I have attached the link for Uncle Vanya (via Netflix)