This is all a very timely discussion. I've just started mandolin lessons and last night we were working on Congress Reel which is written with one sharp and the root note is A. That makes it "A Dorian" and I asked about the harmony to be used with that melody.
The answer was to do A+E chords for the same reason Randy describes. The tune itself is neither major or minor (though if I had to pick one it would be A-minor before A-major, certainly) so don't put the 3rd degree in the accompanyment at all. I learned a neat rhythm pattern alteranting between A+E and G+D notes on the wound strings along with open A and E strings. It's pretty cool stuff, for a beginner it's sort of a brain-full!
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