This will only help find what’s both new and widely popular, but a few times I’ve done something like use the Oricon top singles list as a basis for what to check out. I’ve also seen torrents that will compile an entire such list into a download, though obviously that’s a liiiiittle less legal.

As Zach said, Last.fm is a decent place to go. You can search for tags, such as ‘japanese‘.
There are some good playlists on 8tracks.
There are also blogs out there for Japanese music, such as Sparkplugged.

And if you want to get weird, look up Vocaloids.

Despite all these sources, i still find it difficult to find music that suits my tastes. You may have more success, though.

You should check out

http://newschoolkaidan.com/

and

http://bsgsbkhts.com/

Tokyo Jihen

“Mass of Fermenting Dregs” – Who the hell comes up with names like these? To be honest, I was actually expecting them to be death metal; the name would fit well alongside the likes of “Cradle of Filth”, “Autopsy Torment”, “Gutworm”, and “Cattle Decapitation” :P

OT: As Zach suggested, last.fm is perfect for what you’re wanting. Just go to http://www.last.fm/listen/globaltags/japanese and it’ll play a random selection of Japanese music. I haven’t been there in a while so I just noticed they use a video player now instead of just a simple audio player like before; not sure whether that means they have a reduced selection now, but I’m sure it’s still good.

Also, there should be a thread on this forum called “What I’m listening to now” that has a ton of music suggestions in it. Though good luck finding it, as nobody’s posted to it in a while, I think, and the search feature here doesn’t work properly.
…through the magic of Google, here you go: http://www.textfugu.com/bb/topic/what-are-you-listening-to-now/

On the old version of this forum, there was another thread with the same title, and in that thread I found the very first (and practically only) Japanese group I ever fell in love with: Perfume. In the 3 or so years I’ve been learning Japanese, I’ve only ever regularly listened to Perfume (and MEG, who’s produced by the same guy). Here are the first two songs I heard that got me hooked:

I got hooked on Japanese punk/rock back in the 1980′s, so I can make some general suggestions:
1) The Blue Hearts – cannot go wrong with the seminal Japanese punk band
2) Mongol 800 – sort of powerpunk, but you can really make out the lyrics, and pretty much everyone has heard their hit “Chiisana Koi No Uta.”
3) Toyko Ska Paradise Orchestra – grandaddies of Japanese Ska, going strong since the 1980′s
4) 10-Feet – more power punk
5) The Street Beats – a more traditional punk/rock sound
6) Unicorn – the legendary Tamio Okuda!
7) Sambomaster – more of a pop rock sound
8) Dragon Ash – Japanese Nu Metal
9) Thee Michelle Gun Elephant – Garage rock
10) PUFFY – I have always had a soft spot for PUFFY(Ami Yumi)

You can find most of these on You Tube, and with a downloader/converter program (found free pretty over at place like CNET) you can burn your own CDs Hope that helps some!