The West Wing!

Well, The Wire. The violence isn't the focus and you don't actually see a lot of it.

Also, I'd strongly suggest getting to the end of The Sopranos if you didn't, because it is actually a through plotted and sophisticated morality story.

Seconding The West Wing too.

Oh, and here's one on my list that I haven't seen but which got wonderful reviews and is definitely period stuff: the John Adams mini series.

Showtime has some fantastic period series, notably Rome and The Tudors They're both available via Netflix.

be prepared for LOTS of teh booty in both, tho *groucho marx eyebrows*

I have a very similar taste in TV shows, so here goes:
Lie To Me is fascinating. Tim Roth gets more East-End British at the viewer with each episode ("How ya doing, duck?"; "Don't mither yourself, love.") Which is sort of like watching a vicious ferret act like a snuggly bunny ...
Being Human (the British version) is subversive, funny and (despite the premise) not particularly violent. It draws you in with the characters, like Six Feet Under. You can see this on BBC America or rent the DVDs from Netflix. I didn't like the US version, which seemed to lack the subtle humor of the original. Possibly because I am British(!).
If you haven't seen Dead Like Me, this is a wonderful series -- you are glued to the screen, waiting to see how the next person will die (not at all gory or horrible, but graveyard humor funny).
Midsomer Murders is a show that has been going for about 12 years in the UK. You can get the DVDs on Netflix -- wonderful British mystery writing, the murders tend to take place off screen and the detective investigations are fascinating, with brilliant characterization. If you like this sort of understated detective show, you could also look at Inspector Morse mysteries and the Inspector Lewis DVDs (a spinoff from Morse). All three series are well-written, with a unique backdrop of English villages and eccentricity.

As for period pieces, I'd second Rome and The Tudors. Both European productions, so they are very well done (even if The Tudors does take huge liberties with history). Try The Young Victoria if you want something to stream from Netflix; this is an engaging miniseries. The Miss Marple mysteries from Netflix are a good watch - set in the 1950s or thereabout, they are quietly intriguing.

But my very favorite is the 3-series of Slings and Arrows. I never understood why this wasn't taken up by a major network (it came to rest in the backwater of cable in the USA). I *loved* this show: funny, engaging characters, a shipwreck of a production that is always "alright on the night" (if you discount the backstage drama) and wonderful writing. Available on Netflix.

Three more British series that you may not have seen over here (first two are old, but still very watchable and well written):

Rumpole of the Bailey - the trials and tribulations of a barrister in London's criminal court.
Kavanagh QC - more barrister action, with a strong ensemble cast.
Primeval - dinosaurs come through rifts in time. Worth watching if only for the brilliant Juliet Aubrey in the first three series.

All available on Netflix.

Oh, MAN, yes, Slings and Arrows. I'm so peeved at the scarcity of its episodes. (I really liked the structure, though, of centering each season around a single production.)

How about some of the USA shows? We love Psych for its zingy writing and constant references (not TOO pop-cultury, thankfully), and the cases are more or less a younger and cuter male version of a cozy. People are always recommending Burn Notice, too.

Foyle's War, perhaps, under that BBC aegis? I've only seen a couple episodes, but they were very late.

Also: the endless run of Upstairs, Downstairs.

Exploring the "you might like this" stuff in Netflix has led me to tons of stuff I never knew existed. Lots of late-90s period four-parters, ostensibly after the success of Pride & Prejudice -- maybe not quite as fabulous as that one, but certainly still well-made and easy to watch in hour-long increments. I just finished Wives and Daughters.

Breaking Bad. It's not exactly a period piece, but the mood setting and writing are pretty damn amazing.

Oh, and speaking of Rome, here's a great oldie but goodie miniseries you'd likely enjoy: I, Claudius.

Re Lie to Me, I second, but I have to say I thought it jumped the shark after it's outstanding first season.

Also, speaking of Rome, I have been enjoying Yale's course on Roman architecture. YouTube counts as TV, right?

Stanford's Fourier Transforms for something deeper.

shows where everyone in it is a horrible human being

And you like Mad Men? In our little Mad-Men-watching group, we call it Assholes Doing Bad Things. But we still love it.

It's unrelated to your list, but Sports Night was an awesome show. I recently heard that it was inspired by Keith Olberman's sports show. Cast is excellent, and Aaron Sorkin, who can do no wrong, created it.

Just to echo, because it's worth it: Slings and Arrows is extraordinarily good --- not just the acting and the humor and the stories (though they are all EXTRAORDINARILY GOOD), but the writing.

It's set in a (fictional) modern-day Shakespearean theater housed in a larger theaterical complex. Each series takes place over a theatrical season, and each theatrical season features a different play (Hamlet, Macbeth, King Lear, and a few delicious glimpses of a parallel production of Romeo and Juliet).

And for many viewers, that's all they'll notice, and it is PLENTY: dramatic and funny and touching and emotional.

But each theatrical season also echoes the action of the respective plays, without being at all showy or predictable. It's fantastic writing --- the kind of amazingly well-crafted story that doesn't even call attention to itself, because it doesn't have to.