An old thread, but I feel compelled to comment, as country & western/western swing walking bass is my specialty.
An understanding of the music will help know when to walk and how to walk. If you're talking about Ray Price, Faron Young, Johnny Bush, etc., that style is a shuffle. Shuffle music is a descendant of western swing, which is a descendant of early jazz and swing. In other words, shuffle-style music is DANCE music. Ray Price played DANCES, as is the tradition in Texas.
SO... if your bassman plays 1-3-5-3-1-3-5-3-1 all night, that doesn't go anywhere musically. It's laborious and heavy, and even more so when the drummer hits the kickdrum 4/4 all night. That's ok for modern Nashville music, I guess, but that is way too heavy for dancing. The bass should, when possible/practical, walk it all the way up, i.e. 1-3-5-6-8(1)-6-5-3-1. That lifted style of line extends the phrase, lets it breathe, and lightens it up, as dance music/jazz/swing should never be heavy-laden and laborious. Also, the kickdrum should be 2/4. Piano left hand should mimic bassman, and right hand should comp on the "&" of every beat (1 & 2 &...). Piano left hand should never compete with the bass, only compliment it. With this light, free style of rhythm, the lead instruments can float on top unencumbered by heavy-laden rhythms. It is very freeing to a lead player or a vocalist. Most of all, it is easy for dancers to dance. They don't come to a dancehall to plod, they come to dance. How you play can directly affect how light on their feet they can be.
If you want to play a 2/4 walking bass instead of stacked bass (1-5-1-5), that is a great style when appropriately applied. Used in western swing. Still has a light, moving feel.
Either way, please don't play 1-3-5-3-1 every song all night long. That is bland, boring, uneventful, and doesn't fully allow the music to float or develop. If you're going to listen to Nashville recordings of walking bass/shuffles, they need to pre-date 1975. If you want something current, you might find a rare recording that features proper shuffle playing, but you'll have better luck checking out Jake Hooker or catching us live somewhere. Jake's recordings are great examples. There may be other players out there doing it, but maybe not so prominently. Definitely some guys in Nashville that know and understand, but don't have opportunities like we have here in Texas. Regardless, listening to fine examples is the key.