IJAG, I am a hiring editor, and you're getting some pretty accurate advice from the previous posters. But I'll share my point of view ... most of which will echo what you've already heard.
I do want a cover letter. It tells me that you're paying attention and applying for a specific job rather than spraying and praying. I don't need much, though. A basic, four-paragraph format works just fine.
1st graf: Tell me which job you're applying for (because I might have several unique openings) and if someone I know recommended you apply for it. ("Our mutual friend, John Smith, forwarded your contact information to me about your opening for Executive Agate Editor. I'm interested in discussing this opening with you.")
2nd graf: Tell me why I should look at your resume. In other words, tell me how you can help my company. Don't rehash your resume, but do sell yourself a little bit. And remember that you're telling me what you bring to the table ("I can help you ...") rather than what I can do for you ("I'm looking to do something different with my life ...").
3rd graf: You can elaborate a little bit more on the second graf, if you'd like. An relevant anecdote or short story -- but not your resume in narrative form -- is good, too. This graf is optional, as long as your first and second grafs leave me with the idea that you are applying specifically for this job and you understand what I'm looking for.
4th (and final) graf: Wrap it up. Let me know if you've enclosed/attached anything (resume, clips, etc.). Tell me when you're available and I can best reach you. And thank me for my time and consideration.
That's really all any hiring editor needs today.
I don't recommend it, because attention spans are very short these days, but if you decide to go rogue, make sure it's damned compelling. I once sent a cover letter to a very respected sports editor that opened with one sentence -- "Emmitt Smith wanted me dead." -- and then proceeded to tell a story about how I once broke a story Smith was not at all happy with. The last couple of grafs were more conventional, tying the Smith anecdote to the job opening. He called to say that he couldn't fly me in because they wanted to hire locally, but just wanted to congratulate me on a great cover letter. (Honest.) Given today's job market, I'm not sure I'd take that kind of risk again.
Good luck to you, IJAG.