I have an old megapost of Great UK Drama Series - operating on the premise that a UK-length drama would probably fit the 'miniseries' mould you're after! - that I'll dig out:
Life on Mars [IMDB: 8.3] (Plus the followup series, Ashes to Ashes [IMDB: 8.2])
Hustle [IMDB: 8.2]
Spooks [IMDB: 8.3] (MI:5 over in the US).
Cracker [IMDB: 8.5]
House of Cards [IMDB: 8.6] (The original. Plus the followup series, To Play The King [IMDB: 8.5] and The Final Cut [IMDB: 8.2])
GBH [IMDB: 8.8]
The Singing Detective [IMDB: 8.8]
Our Friends In The North [IMDB: 8.7]
This Life [IMDB: 8.6]
Auf Wiedersehen Pet [IMDB: 8.6]
Boys from the Blackstuff [IMDB: 8.7]
Takin' Over The Asylum [IMDB: 8.9]
Edge of Darkness [IMDB: 8.6]
Talking Heads [IMDB: 8.6] (Talking Heads 2: [IMDB: 8.7])
Messiah [IMDB: 8.0]
State of Play [IMDB: 8.5]
Call the Midwife [IMDB: 8.4] (Not for everyone, of course, but it's really not as twee as the premise sounds!)
Sherlock [IMDB: 9.2]
Happy Valley [IMDB: 8.5]
Line of Duty [IMDB: 8.4]
The Night Manager [IMDB: 8.2]
Black Mirror [IMDB: 8.9]
Inside No. 9 [IMDB: 8.4] (An anthology series, so somewhat patchy, but the second episode of series 2 - The Twelve Days Of Christine - is phenomenal [IMDB: 9.3])
Being Human [IMDB: 7.8]
In The Flesh [IMDB: 8.1]
The Wrong Mans [IMDB: 7.9] (Strictly speaking, it's a comedy, but it's a parody of thrillers that is, in turn, actually quite a good thriller in itself)
Taboo [IMDB: 8.9]
Broadchurch [IMDB: 8.4]
Death in Paradise [IMDB: 7.9] (Light detective series set on a carribean island. Not terribly gritty, but good escapism)
The Missing [IMDB: 8.2]
Luther [IMDB: 8.6]
I haven't yet got into the Scandi-drama craze that's going on a lot here, but the big names I've heard of in that regard:
The Killing [IMDB: 8.4]
The Bridge [IMDB: 8.6]
Borgen [IMDB: 8.5]
In terms of ones from that that are currently a going concern, Line of Duty has just ended its fifth series, and while I've not yet got into it myself, seems to consistently get huge buzz whenever it's on.
Things that are newer than when I made this list that are worthy of note:
Fleabag
Killing Eve
Maybe Gentleman Jack (It's only just begun but it's getting pretty good word of mouth)
Maybe Years and Years (Similar)
The Bodyguard
Baptiste (The follow-up to The Missing)
I can't actually vouch for all of these directly, but they're all series that clearly caught peoples' imagination when they were broadcast.