I've been trying to train myself to hear the JI resonances (Harmonic Experience has tons of exercises for this). I can hear the sharpness of the ET major third and the flatness of the ET minor third fairly easily.
Barbershop quartet music (and other a capella music) is a great place to hear just intonation. The consonant barbershop seventh chord was a relevation to me when I first learned of it. It's really close to the dissonant dominant 7th interval, but serves a very different purpose harmonically. The minor seventh is also a different interval. Thus there are actually 3 seventh intervals, all of which are approximated in ET by a single note. This blog post has a good summary:
Barbershop quartet music (and other a capella music) is a great place to hear just intonation. The consonant barbershop seventh chord was a relevation to me when I first learned of it. It's really close to the dissonant dominant 7th interval, but serves a very different purpose harmonically. The minor seventh is also a different interval. Thus there are actually 3 seventh intervals, all of which are approximated in ET by a single note. This blog post has a good summary:
http://www.garygarrett.me/?p=1575