If you're referring to the harmonies sung by the BGVs (background vocals), then they will follow the harmony (chords) of the song. Yes, you're partially right about 3rds, but 4ths are used as well. It really depends on the chord that is occurring at any given time. I assume most country songs are very tonal (sticking to a major key or a minor key) chords are comprised of 3rds and/or 4ths depending on the inversion used.
For instance, let's say the song is in the key of C major and the chords are F major, C major, G major and A minor (4 beats each). The notes for F major are FAC, C major is CEG, G major is GBD and A minor is ACE. The harmonies sung by the BGVs for F major will be mainly F, A and C. The same is true for the other chords...they will use notes from the chord. This is an overly simplistic view of harmony. F major has three forms: root position (FAC) which is all 3rds, 1st inversion (ACF) which is a 3rd and a 4th, and 2nd inversion (CFA) which is a 4th and a 3rd.
Polyphony is used widely when it comes to harmonies for BGVs. Let's stick with the chords from the previous example and introduce a melody. Let's say the melody for the first bar is F G A G A (remember the first chord is F major). The BGV harmonies are not just simply going to stick with notes from the F major chord. They're going to move up and down accordingly with the melody. This means that there will be other chords introduced when the melody note is not on a chord note. In this example, any time the melody hits a G, it's not part of the F major chord. What notes will the BGVs sing then? Since we're in the key of C and the melody note is a G, we can basically use any chord in that key that has a G in it. This means, we can use C major (CEG), E minor (EGB) or G major (GBD). The chord that the BGVs use is subjective because they will all work, but to each person, they will have a preference.