In Ionian, the primary triads are [I], [IV], and [V] - just like you've mentioned.
In Dorian, the ^6 degree brings out the Dorian flavor so the [IV] clearly stands out, and I've gone through [i]-[IV]-[VII]-[i] chord progressions in the mode quite often - so in Dorian, I would say the primary triads are [i], [IV], and [VII].
In Phrygian, notice the minor [iv] chord exists. The [II] is usually "substituted" for [vo], and as I read JdJazz's answer on Harmony in Phrygian : What chords are tonic, subdominant, and dominant?, I'd go with [i], [iv], and [II].
In Lydian, I could think of [I]-[II]-[V]-[I] progressions very strongly, so in this view, it would be [I], [II], and [V].
In Mixolydian, we have a major [IV] chord, and because of the minor [v], the [VII] chord is often "substituted" as a dominant, resulting in [I], [IV], and [VII].
In Aeolian, as Michael Curtis had mentioned, we can permute the , , and  chords here. So it's obviously [i], [iv], and [v].
In Locrian, User16935 had mentioned about truncating the tonic chord to just root and third on Locrian Harmony. Due to the flattened [V] chord, I would say the [iii] acts often like a dominant "substitute", and we also got a minor [iv] chord - So I'd go with [i(no5)], [iv], and [iii].