I’m interested in musicians/bands/songwriters whose songs evidence a strong connection with a city/state/country.
Some examples would be Bruce Springsteen (Jersey Shore), Dropkick Murphys (Boston) and Gordon Lightfoot (Western Canada). It’s not just that they are from those places, it’s that their songs are very often located in those places, either explicitly, or just in the imagery they use (e.g. Springsteen boardwalks and factories, Lightfoot snow and wheat).
What are some more?
Dr. John.
I don’t think the Ramones could be from anywhere but New York, and it shows up a lot in their songs.
Primus makes a lot of references to San Francisco Bay Area towns and locations.
Great Big Sea?
You beat me to them, Helena.
Lyle Lovett and Texas
Kinks: England
Mellencamp
Sublime mentions Long Beach, CA in their songs, for example:
from “Summertime”
*local slang for the city of Long Beach, which I hadn’t heard before even though I live near there. :o
Red Hot Chili Peppers and California.
John Denver and Colorado.
The Pogues and Ireland.
The Clash, Sex Pistols and England.
Bon Jovi and New Jersey.
Lucinda Williams and the ArkLaTex (she’s from Arkansas but sings about East Texas and Louisiana a lot)
Lou Reed and New York City.
Dieselhed and rural Northern California.
If he’d stayed in the Midwest, I’d nominate Sufjan Stevens for Greetings from Michigan and Come On Feel the Illinoise, but he’s a Brooklynite now and will certainly be releasing the other 48 CDs in the series any day now.
Oddly, the first person I thought of when reading the thread was Astor Piazzolla, whose music seems like the essence of Argentina to me, although I’ve never been there and there are no lyrics to his music as far as I know.
I think you mean LBC, not LBZ, unless that was just a typo.
Billy Joel - Long Island (NY)
The Beach Boys - Southern California
Or the Long Beach Combat Zone. 
Texas: Willie Nelson, Robert Earl Keen, Pat Green, Stevie Ray Vaughan, the Butthole Surfers.
Craig Finn from the Hold Steady writes about the Twin Cities pretty much exclusively.
ZZ Top… that little ol’ band from Texas.
Stevie “Guitar” Miller has long held ties to Dallas.
Kate Bush’s love for England, and identification with her home is one of the great love stories of all time. Her recent album Aerial is a two CD set, and the second disc A Sky of Honey is an album length song suite about one perfect day in the English countryside.