A method that I have found to work very well when trying to figure out suitable chords for backing up an Irish tune ("not song !" Did I say that ? Well why not for songs. Try any of the songs on any Altan CD. Their guitarist uses DADGAD, and it works for the songs also.) is to think in terms of chord scales, i.e. find a suitable chord for each note in that scale. This gives you a lot more possibilities than the often-used I-IV-V chord progression, and is less obtrusive than trying to "jazz it up" with all kinds of complex chords. And it almost naturally results in (very professional sounding) ascending or descending bass lines.
A "system" that often results in something useful is the following:
I IIm7 1st inversion of I IV V VIm (or 1st inv. of IV) 1st inv. of V I(octave up)
A few examples are given below.
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Of course these examples are valid for the major scales of D and G only. For the minor keys, and also for the modal scales, you can use a similar approach.
Here are two more examples, a low and a high Dm scale. You can work out others for yourself if you have read my piece on the theory behind chord scales.
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