Can I use the term "tweets" for posts at my own site? And if I put the word "tweet" in quotes?
From what I can see on the USPTO registry, "tweet" is still not trademarked for the purposes you're talking about.
Twitter has been working to secure that trademark, but I can't find any record of them having actually secured it.
Just the same, I'd predict that using it will get you at least a cease-and-desist letter, and possibly a lawsuit. Whether you'd win that case seems to be a very open question, but that presumes you have the resources to litigate.