Hi GTR88 and welcome on board !
One of the most famous hotel with japanese rooms (tatami) is the
Meguro Gajoen, in Meguro:
http://www.megurogajoen.co.jp/english/guest.html
You also have some western hotel with japanese rooms, like the
Four Seasons Chinzan-So, but the price will be expensive (suites).
If you're not looking for overpriced rooms, but reasonable prices, you can try some ryokan inside Tokyo:
Tokyo Ryokan, in Asakusa.
Ryokan Kangetsu
Sawanoya Ryokan
And
Andon Ryokan, if you're not against some "modernism" in the rooms.
Hope this helps !