I keep hearing popular country songs where the bass is all fat rich tone but NO character, like string noise or edge or anything. How do they do that? I know it sits well in the mix without interfering with anything else, but it seems odd to a bass player.

No string noise can be a product of good technique and string choice  However there is also ;_____Signal processing___ post the players control in some case's ( Engineering and producers choice). EQ , Compression , noise gates  , etc... ...  I have heard of many Nashville studio tricks for everything from bringing out or reducing the sibalance in a Female vocal to Recording Bass as per your description.

Compression is an essential tool in making sure the bass is consistent in recordings, and very preferable for live performances.  Generally, the more you compress, the less character you'll hear.  Playing style and equipment is a big part also.  One thing to add is that there are a few standard bassists that work in Nashville that end up playing bass on many, many albums of various recording artists, so the consistency in tone may have something to do with that as well.  (Same player - same tone).

Processing.  I'm convinced there is a Pro-Tools plug-in that specifically extracts the soul from the bass track.

Definately compression.  They will compress the bass track several times in the process.  I know for a fact that when laying down the bass track they compress it twice.  That is before anything else happens to it.  Then of course more compression later on.

Another issue here is that there is a lot of tuning down, it seems that 5 string basses are frowned upon in Nashville.  I tried it for awhile with my Epic and it really does change the character of the bass. It can be a tad funky to get your head around that your notes are in a different position then that the other players.
Also I agree with stout71 that there seems to be only a few players on all those tracks, maybe not so much the tone, but, the approach.

Spelling correction to my post #2118   ;  SIBILANCE instead of SIBALANCE.

Well, you've got Glenn Worf, Dave Hungate, Lee Sklar and a handful of others, and we've all heard their work elsewhere sound completely different than the Nashville stuff.  Lots of 5-string now as well.  It's interesting to me that you can't even tell if they're using rounds or flats, or often which octave they're really playing.  Just have to keep in mind that Country Music as a genre doesn't have much room for bass out front, what with twangy guitars, fiddles and steel..
 
John
 
(Message edited by 811952 on November 03, 2011)

SIBILANCE:
 
Of, characterized by, or producing a hissing sound like that of (s) or (sh): the sibilant consonants; a sibilant bird call.
 
SIBALANCE:
 
how a Series I Alembic instrument hangs on the player's body:  my SI balances well, no neck dive
 
 
 
couldn't resist.
 
Bill, tgo

That's A good one Bill !        I like the way that my SI  81-1940 hangs on me too !

Back  starting in the early 1980s I was in a few Country Bands in the San Francisco Bay Area to about 1997. In one of those Bands( The Bounty Band) The  Original Drummer and I started breaking some of the unwritten rules of traditional Country Music when we started to syncopated the rhythms , and make it a bit more funky. We were both reprimanded by our fearless leader and frontman lead singer rhythm guitar player and told to CUT IT OUT , you guys !!!____at a live gig,  Well, __ Just as soon as he finished giving us a piece of his mind the Owner /proprietor of the venue comes up to the stage and said . That's not like any Country I have ever heard , BUT , They are dancing and jumping around like JACK RABBITS on the dance floor and buying drinks and getting hungry and even buying snacks, DON'T CHANGE ANYTHING   That is what he told us.  A few more nights  the same as that and  he asked us for an extended gig that lasted several years as a house band. The Guitar players followed me and the drummer by playing a bit more less tradition country style and more rock style and tone, We had a few great Pedal steel players,   one of whom played a MEAN LAP STEEL WITH FEEDBACK !   That Band lasted about 14 years more or less in various states of personnel while I was involved. We once opened for  Johnny Paycheck.   Country Music was changing , The  Bakersfield Sound in Country and Outlaw Country was on the rise Breaking traditional sounds and styles in multiple  parameters.  And ________it was fun doing it !  Except for the Gig where we had to play behind the CHICKEN WIRE  covered stage!  YES __ those places really existed______.

We did country like this:
 
That was the mid/late-90's.  I think that album was a Modulus Q4 with Rotosounds direct into a Summit Audio TPA-200B.  I'm quite positive about the 200B, not 100% sure about the rest.

Briant - cool tune!!!
 
I'm from Nashville and not too many folks I know like country music.  The guys I know do it for money (at least the few I know that play for $$$).  Growing up, we kind of made fun of it.  And BTW - HeeHaw ain't real   LOL
 
But I do like me some picking and grinnin  )
 
Stephen