My preference is for hymns with melodies having strong chordal notes - mainly major chords, without related key chords. Several hymns mentioned above and other that come to mind have these melodies:

I'll Fly Away
Abide with Me
Amazing Grace
Sweet Hour of Prayer
He walks with me and he talks with me
How Great Thou Art
My God is Real
Roll Jordon Roll
Ava Maria

Will Your Anchor Hold has a section in the chorus where it runs down the notes of the chord without any inbetween notes.

Memorable folksongs, like The Water is Wide and Tom Dooley, also have strong chordal melodies, as do many country songs. Some momentarily diverge from the three major chords before returning, like Alan Jackson's beautiful song, Here In The Real World.