I find Captain Birdseye's comment rather confusing, when he suggests Fmajor is a good chord to use in A dorian tunes. Since there is no F natural in the A Dorian mode, I think almost any use of an Fmajor might jar and confuse the ear, even if it fits with a melody note (eg an A or a C).
Cap'n, I might be talking total rubbish here. Can you link to a recording of an A dorian tune with an F chord in the accompaniment, so that we can judge what you are suggesting?