Just like my preference for any music in English, my taste in Japanese music is also pretty eclectic.
The Blue Hearts (Japanese 80s band. Wrote a bunch of really catchy songs, and recently the movie "Linda Linda Linda" was based around their song "Linda Linda.")
Moriyama Naotarou (More of your kinda emotional music. I was introduced to his song "Sakura" by one of my Japanese teachers once and I've grown to like many of his other songs too. Btw, there are way too many Japanese songs called "Sakura."
Sakamoto Kyu (Ooold song, but still a classic. In the states (not sure about the rest of the world) his song is called "Sukiyaki" but really its called "Ue wo muite arukou" (something along the lines of "Walk and gaze up"))
High and Mighty Color (much more modern and "rocky" than my previous listings, but I really like their "sound", as cliche of a word as that is. First got into them from hearing "Ichirin no Hana" as it was one of the opening themes to Bleach.)
Elle Garden (Was introduced to this by one of my friends, and whether or not you can really think of most of it as Japanese music (most of the songs are in English), it is a Japanese band, it is catchy, and I like them! One of my favorite songs of theirs would be "Good Morning Kids" on the Pepperoni Quatro album.)
Nobody Knows+ (I've never really been too much into rap, but damn, these guys are catchy as all hell. Especially the song "Kokoro Odoru.")
Service Ace (If these guys ever actually make it, watch for them! They're a band I met when my friend and I went to a livehouse performance in Japan once. If you don't know, a live house is where a bunch of usually unknown bands get together and perform. Although from a quick net search it looks like they may be picking up a bit, cause it looks like they might have a record deal. Anyways, a good song is "Bad News Life."