wow, went through my J-Dorama phase years ago. The only ones that stick out in my mind right now are Long Vacation and Hero both starring Takuya Kimura a member of SMAP! (yes, cool name right? ) I'm not a Kimutaku or anything but I did enjoy his underdog persona in Hero, except his poofy orange jacket annoyed the hell outta me. Long Vacation is just a classic and I'll leave it at that, good shite. If you really want some melodrama I suggest checking out some K-Drama, Koreans love to make themselves cry for no reason.
I've seen a few others, Gokusen, Orange Days (Road?), etc. but can't say that they made much of an impression. Then again I've become old and boring recently so take my opinion with a grain of salt.
edit: How could I forget...the GTO J-drama is excellent.