The Bass Notes
Backup guitar for old-time fiddle music basically consists of hitting the bass note of the chord then strumming. The bass note is the name of the chord (the root) played on the 6th, 5th or 4th string. An alternate bass note is another note in the chord besides the root played on one of the bass strings. For example, a D chord consists of D-A-D-F# starting with the 4th string. The D note on the 4th string is the bass and the A note on the 5th string would be an alternate bass. So, when you are playing a D chord you can alternate between the bass D and the alternate bass, A. The F# on the 6th string would also be considered an alternate.

G chord: G-B-D-G-B-G.  The G on the 6th string is the bass and the B on the 5th and D on the 4th are alternate bass notes.

C chord: x-C-E-G-C-E.  The C on the 5th string is the bass and the E on the 4th is the alternate. The E and G on the 6th would also be considered alternate bass notes

A chord: x-A-E-A-C#-E.  The A on the 5th is the bass and the E on the 4th is the alternate as well as the open 6th string, E.

E chord: E-B-E-G#-B-E. The E on the 6th is the bass. The B on the 5th is the alternate.

Notes on the guitar:

Open Strings  _E__A__D__G__B__E_
    1st Fret   F  Bb Eb G# C  F
    2nd Fret   F# B  E  A  C# F#
    3rd Fret   G  C  F  Bb D  G
    4th Fret   G# C# F# B  D# G# 

A few chords showing the alternate bass notes:

   D Chord       G Chord      A Chord
__A_D______   ____D_G_B__   E_A_______E
| | | | | |   | | | | | |   | | | | | |
F#| | A | F#  | B | | | |   | | E A C#|
| | | | D |   G | | | | G   | | | | | |
| | | | | |   | | | | | |   | | | | | |

Listen to this example: Whiskey Before Breakfast.
Part A
D---|D---|G-D-|A---|D---|D---|G-D-|A-D-:||
Part B
D---|D---|A---|A---|D-A-|G-D-|G-D-|A-D-:|| 

Bass Runs
When changing chords you can do a little bass run up or down to the next chord. If  you're going from an A chord to a D chord you can hit the A bass on the 5th string then, instead of strumming the A chord, you play the C# on the 5th string on the way up to the D bass on the 4th string. There are many ways to do a bass run. You can play two bass notes in one beat or leave out a strum to get another bass note in. You can just play the notes in the scale or play all of them. You'll want to end up playing the root of the new chord at the end of the run. Below are some possibilities.

From D chord to A chord.  The bass run may be D, C#, A.
From D chord to G chord.  The bass run may be D, F# on 6th string, G.
From G chord to C chord.  The bass run may be A, B, C.
From G chord to D chord.  The bass run may be B, C, C#, D.
From C chord to G chord.  The bass run may be C, B, A, G.
From E chord to A chord.  The bass run may be F#, G#, A.
From D chord to E chord.  The bass run may be D, A, E. You can also play D, D#, E.
From C chord to G chord.  The bass run may be C, B, A, G.

As you experiment remember that some bass runs will sound good or not depending on what the other musicians are doing.

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