How's the prequel trilogy?
I am probably one of the few who actually prefers the prequel trilogy as a whole compared to the original trilogy, so I guess there will be some bias in my answer, but I can try to explain what it has going for it:
Unlike the original trilogy, the prequel trilogy feels more connected with each passing game. There are more overarching/unifying mysteries that are built up with each passing game (and movie since Eternal Diva is part of the trilogy). Some fans says the prequel trilogy feels like one large game because of it. I'd say this is easily the prequel trilogy's strongest point. The cliffhangers and mysteries left behind as each game finishes will make you want to jump straight into the next one.
Another highlight of the prequel trilogy is Jean Descole. As Don Paolo was to the original trilogy, Jean Descole is to the prequel trilogy: that is, the trilogy's main antagonist. Unlike the antagonistic yet rather comical Don Paolo, Jean Descole is much more sinister and mysterious villain in his demeanor and goals. He also makes a very good foil to our favorite Professor. Both act calm most of the time, but Layton will know to keep his cool when things don't go as planned, whereas Descole is more prone to anger when things don't get his way. Both are masterful inventors, but whereas Layton uses his inventions for good, Descole uses his inventions for evil (as well as to further his goals). You learn more about Descole's character with each passing game, but I'd say it's the movie where he shines the most.
The prequel trilogy also does a solid job fleshing out the supporting cast with each game, to the point you can feel how integral they are to the given game's story. Arianna, Randal, and Henry are some of my favorite characters from the prequel trilogy because of the spotlight they are given.
I guess the main issue with the prequel trilogy lies in the way it all pays off by the last game. You can feel the developers were definitely being ambitious by Azran Legacy, but the way how some of plot elements/mysteries pay off don't feel as grand as they were made out to be. It's a bit hard to explain without going into spoilers, so I'll just leave it that.
In all, I definitely would recommend playing the prequel trilogy because it has its fair share of great characters, an intriguing villain, and some very memorable plot twists. Be sure you watch the Layton movie, Eternal Diva, after finishing Last Specter too, since Miracle Mask actually references some of the key events that happen in the movie. The movie itself is pretty good for reasons others have already mentioned.