Watch the Mr. Chester Breakdown 2/3 online guitar lesson by Muriel Anderson from Innovations for Acoustic Guitar
I'll describe my way of getting a crisp alternating bass sound for the tune "Mr Chester," using a right hand technique that is actually derived more from Leo Kottke than from Chet. Having said that, it's also cool to play this tune with the traditional Chet Atkins / Merle Travis muffled bass, anchoring the palm of the hand near the bridge. You may want to use a thumb pick if you play it this way. In either case, don't be afraid to use the weight of your arm to get a nice solid bass sound, either with a thumb pick or without. For a really driving bass in the style of Merle Travis, check out Thom Bresh's
Whenever possible, keep the right hand in a standard position by playing the first string with the ring finger, second string with the middle finger, third string with the index finger, and the thumb gets the three bass strings. I'll work through the whole tune here.