One of the more typical approaches to rockabilly rhythm guitar is a combination of the low end of an open chord and a walking bass figure. Generally the root, fifth, and octave are a good choice for a basic chord voicing; then a simple pattern is superimposed on top of the basic chord.
In Example 1, we're using a basic root/fifth two-note chord as our foundation, but we are applying a simple arpeggio picking approach. That is, you would play the root (E), then the fifth ( B), on the A string, then the root, then the sixth (C#) on the A string. Make sure the root note sustains so that the added notes on the A string harmonize against it. This is your basic blues/rock rhythm figure, one near and dear to most guitarists. It is also great for a basic rockabilly rhythm figure.
In Example 2, we mutate the same figure to travel a bit further up the Astring to the seventh (D), so that you are now playing root/fifth, root/sixth, root/seventh, root/sixth, and repeat. Think "Honky Tonk" by The Bill Doggett Trio. This lick works exactly the same way in the key of A, using the root and fifth of an open A chord. Then, by changing your fingering to a basic root/five barre chord and adding the sixth and seventh with your pinky you can play this figure on the B chord, and throughout a twelve bar blues form. If it sounds overtly bluesy to your ears, I would point out that the primary difference between rockabilly and the blues is largely one of feel and approach, rather than harmonic content.
Now, if we syncopate a walking bass line inside a bit more of the E chord (root/fifth/octave now), you'll hear a more proto-typical rockabilly rhythm start to take shape. Try playing Example 3 with a snappy eighth-note shuffle feel: the root on the low E string, then the fifth and octave played together against the root; the third (G#) at the 4th fret on the low E string, then the fifth and octave together against the G#; the fifth (B) at the 2nd fret on the fifth string, then the fifth and octave together; then the sixth (C#) at the 4th fret on the fifth string, then the fifth and octave together again. Then repeat the entire process. Basically, what you're doing is playing a I/III/V/VI bass line on the downbeats, and playing a partial E chord on the upbeats. This is one of the quintessential rockabilly rhythm figures. Add a little tape echo or analog delay slapback, and Man, you're real, real gone!