My two cents on what I've played of the series:
Shadow Hearts - Loved it. The only one I've beaten of the three though. It was just a nice game to play through, not particularly difficult ( Yuri being brokenly strong ), different atmosphere to most other rpg's I can think of and the Judgement Ring at least made the battles feel a bit more individual than most games'. Also, the soundtrack was excellent, with many strong tracks throughout the game.
Covenant - A bit meh compared to the first one. I'm only just in the second or third area of the second disc though. As has been said; the battles really had a bit more depth added to them with the attack types, positional consequences and defence. I also liked the little Solomons' Key riddle. I like the new characters as well. But I've been able to put my finger on two things that irk me about the game: 1 - No music when in dungeons, has anyone else noticed the fact that you get that same howling/ambience playing in pretty much every dungeon area? It's a shame to not use the talents of someone as capable as Hirota. 2 - The monsters just aren't as gruesome as the first one, making it feel a bit more generic in my eyes.
From The New World - An improvement over Covenant, but still not reaching the highs of Shadow Hearts. I love the quirky characters included in the game and best of all:
Having Al Capone as an ally was ace!
The musics as good as the first games' soundtrack, and the new setting of the Americas is a fresh change. But I still can't shake the irk that I'm simply not enjoying it as much as Shadow Hearts. Again, more than anything I think it's the atmosphere that it can't quite match. There's also the slight issue that it feels as though it's growing closer to its' end and I'm still not satisfied, I'm waiting for there to be some huge twist that'll extend the games' lifespan somewhat.