Kelly from Georgia identified my unknown bird species: Eastern Kingbird.
That's what my birds are, without a doubt; the markings are correct, right down to the white on the tip of the tail. I found information about them HERE, and it describes their behavior exactly.
Thanks so much, Kelly, and all of you who ventured a guess. There's nothing like the World Wide Web, is there? And nothing like a network of Internet friends to help solve a mystery.