Alright then, lets go with that bass walk over chords.
Lets pick three chords. I choose C F and G, but really, any chord progression will work, so long as you stay within the triadic chords of the key. But for now, lets just stick with old reliable I IV V progressions. In C, that's C F G. For our purposes, we're going to talk about the strings in terms of "basses" and "trebles." The basses are E A and D. The trebles are G B and e.
It's important to know which strings you would normally strike when you strum a chord. We mute the big E for open C, striking only the strings from the A on down. If you use a full barre for your F, then you would strike the E string, but if you're like most humans, you're going to just barre the B and little e string with your F, and strike the A string on down. G is a full chord, so you strike all the strings, starting with the big E.
So go ahead and fret the C chord. Now take your thumb and alternate between striking the A and D strings. Use your thumb. Not your fingers. They'll have their own job to do in a minute.
---------------
---------------
---------------
---2----2-----2---2-
-3---3-----3----3-
x---------------
Now keep that same alternating pattern going, and transition to the F. The tab now looks like this.
---------------
---------------
---------------
---3----3-----3---3-
-3---3-----3----3-
x---------------
Now to the G. Move your alternating rhythm to the E and A strings, though.
---------------
---------------
---------------
---------------
---2----2----2---2-
-3---3----3----3-
So that's the walk part. It's done completely on the basses. Now lets put the rest of your fingers to work.
Move back to C. Start your walk with your thumb, but between each thumb strike, pluck the treble strings with your remaining fingers. The walk then looks like this tabbed out.
----0---0----0----
----1---1----1----
----0---0----0----
------2----------2-
-3---------3-------
x------------------
Then progress through the chord changes as you did before. It's pretty easy with a little practice. I sometimes use a pick on the basses if I want the bass walk to stand out a lot, but using just your thumb is easier, I think. Work on training your treble fingers to find their assigned string naturally. Once you're doing it it will look like your hand is barely moving at all.
If you're comfortable playing scales in position, then this becomes a really powerful tool, too. You can alternate between your bass walk, and your melody line and never come out of position.
So there it is. Walking bass lines made easy. I hope. 