Could the voiceover provided by Carrie in each episode of _Sex and the City_ be the closest ancestor?
It's a technique also used in _The Wonder Years_ and, currently, _Scrubs_.
I was going to mention "The Wonder Years." If I'm not mistaken "MASH" (the TV show) also used voice-over occasionally, with Hawkeye writing a letter home to his father about camp happenings.
The kinds of shows you describe, with a central cast of characters that act as a frame for each week's "cases" are called anthology shows, and medical anthologies in particular have a long and fascinating history. A great book on the subject is Joseph Turow's Playing Doctor. It's out of print nowadays, but you can still find it at places like Powell's and many libraries. It's a fun read.
One problem these shows have is that, while they need to have main characters and interrelationships that are interesting enough to get audiences watching, the weekly storylines and longer sub-plots with guest characters take up a lot of screen time, and so it can be hard to develop the characters fully over time. That's why many of them end up relying on melodrama to keep viewers' interest.
Sorry to go on at such length, but it's a pet research interest of mine. Obviously. 
(Oh, and hi! I've been lurking for a while, and finally found the right opportunity to join the dance.)
Interesting observation about Grey's Anatomy. Of course, it's staring me right there in the face, and I still never realized how many women there are. Although, ER has lots of female doctors too: Abby, Carrie, Susan, Neela. (And there are male nurses, too.)
Serious voiceovers kind of annoy me, although I don't come at it with a literary background. The Scrubs voiceover doesn't bother me, though, because it's supposed to add to the comic value of the show, and most often does that well.
I was going to say that I remember voiceover narration in The Wonder Years, a show I liked a lot as a kid.
i've never watched grey's anatomy, but have been a fan of DH. i think the voice-over narration, after a long absence in the television world, is gaining popularity again.
it's interesting that DH seems to attract a lot of lit people, in my opinion. a lot of my friends and at least one of my lit professors watch the show regularly.
I think Ally McBeal had voiceover narration too.
So does Sex and the City - the Carrie Bradshaw character is usually reading her newspaper column. Other shows with voice-over narration: Malcolm in the Middle - Malcolm is the narrator. Arrested Development - the voice is Ron Howard; the character is Michael Blume, played by Jason Bateman.
oops,sorry, didn't see gzombie's mention of SATC
malcolm is as much talking to the camera as voice over.
yea, you're right. I haven't watched it in a while.
very interesting...i haven't seen most of these other shows -- and thanks, Plin, for the critical terms -- I was sure there was a term for these shows so I'm glad to learn what it is.
Arrested Development also uses narrated VO.
Mel! I found a pretty extensive list:
Magnum PI
The Wonder Years
The Waltons
Dobie Gillis
Doogie Howser
The Bernie Mac Show (direct address)
Titus (direct address)
My So-Called Life
Felicity
Clarissa Explains it All
The Patty Duke Show
Sex and the City
I found all of this in a paper that is available online on pdf. The address is:
http://www.cindyroyal.com/
royal_...l_sex_paper.pdf
Interesting observation, Mel. The "character development over time" strategy is also used with great effect in mystery novels with series characters (often the detective protagonist has some sort of mystery in his/her own past, too, that is slowly revealed, e.g., Sue Grafton's Kinsey Milhone.)
On tv, writers have the additional challenge that sows in syndication may be aired out of order. Writers are often deliberately sketchy on the details of characters' personal lives, histories, in order to allow more flexibility or future shows.
Awhile back, pre-Radway, Tania Modleski wrote a great book on romances & soaps that discusses some of this.
I can relate to the "setting" of a show influencing what I watch... And to put it simply I am a bit addicted to medical shows. I never miss scrubs, house, crossing jordan (which is crossed with crime drama) and have been watching grey's anatomy.
I have never watched er though.
yeah.... tv.
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