I came into the thread to mention Black Sheep Boy and am shocked (shocked!) to see it mentioned already. 
Tremendous album by a band that deserves a lot more recognition than they get.
So, in its place, I’ll offer:
Misplaced Childhood by Marillion,
The Hazards of Love by the Decemberists,
and my current favorite, The Monitor by Titus Andronicus.
I listened to Running on Empty today, in order. It definitely fits the OP.
I wrote about just such an album: Trees Of Mystery in “Mooks On Parade” for my blog! I even explain why I think the album needs to be listened to in order.
There’s a link at the end of the article where you can listen to the whole album, or you can just click here.
Radiohead has been mentioned, I’ll second OK Computer and Kid A.
I’ll also throw in The Avalanches’ *Since I Left You*.
It’s probably less known, but Pitchfork put it at number 10 of their top 200 albums of the 2000s.
From a personal standpoint, this album is sublime. In my top 5 albums of all time. If you’ve never heard it, give it a listen - though, of course, you must listen to the full album in one sitting.
Nine Inch Nails: The Downward Spiral
Camper Van Beethoven: New Roman Times
Hüsker Dü: Zen Arcade
Gotta love the concept album!
XTC’s Skylarking. Although the song selection and order were both heavily influenced by producer Todd Rundgren, the final result is a song cycle of an ordinary man’s life, from childhood through his becoming a widower in old age. A modern classic.
Lots of agreement with what’s been said so far. I’ll add the following:
Elton John, Captain Fantastic and the Brown Dirt Cowboy
Klaatu, Hope
Genesis, The Lamb Lies Down on Broadway
Counting Crows’ “This Desert Life” seems to be built around listening to the whole album. Especially the way it starts out very high energy and seems to wind down towards the end.
Pink Floyd, The Final Cut and Wish You Were Here need to be added to others contributions
Jethro Tull, Thick As A Brick.
U2, Achtung Baby
Layla
Any Grateful Dead concert album, of which there are dozens.
I have it on tape and thus have never listened to it out of order, but the jacket of Lou Reed’s New York specifies it’s been designed to be listened to as a whole story.
Pepperlandgirl, I’ll add Wilco but for me the album is Yankee Hotel Foxtrot (which gets my vote for one of the greatest albums of the 2000s period). Benefits tremendously from album treatment, though the songs can stand on their own.
Some great lo-fi albums that employ thematic and aural segues throughout:
The Microphones - The Glow, Pt. 2
Neutral Milk Hotel - In the Aeroplane Over the Sea
Also:
Modest Mouse - This is a Long Drive for Someone with Nothing to Think About
Lots of (most?) post-rock albums (Explosions in the Sky, Godspeed! You Black Emperor) are really only sensible if listened to in order.