Nishiyama in Kappabashi has a wide array of lacquer dishes as well as affordable vessels for daily use. These red and black bowls in the left box are advertised as something that you can put in the microwave.
These wooden bowls are good for miso soup, but also great for ice cream.
Nishiyama also has a selection of bento boxes, trays, and much more.
Nishiyama Shikki 西山漆器