Today when anyone mentions a Japanese motorcycle, we tend to think of the big four, which are Honda, Yamaha, Kawasaki, and Suzuki. Like every other part of the world, there were other manufacturers, some who are long forgotten. In no particular order, here is the most complete list I could come up with. I suspect that the majority of these firms existed pre war, with the remainder either being swallowed by the big four, gone out of business, or switched to producing a different product. For example, Daihatsu, Mazda, and Mitsubishi switched to automobiles. Another point is that the vast majority of these manufacturer's product was not sold in North America. The list is as follows:
Abe Star, Ace, Aero, Aichi-Kikai, Aioi, Aisan, Akitsu, Amano, Asahi, BF Motor, Bim, Blue Bird, Bm, Bridgestone, Brother, Cabton, Center, Chiyoda, Daihatsu, Fuji, Fujitsubo, Giant, Hirano, Hodaka, Hosk, Hyogo, Iwasaki, Kurogane, Kyoho, Lilac, Marusho, Mazda, Meguro, Mitsubishi, Miyata, Mizushima, Ns, New Era, Nisshin, Rikuo, Shin Meiwa, Showa Fujiya, Tohatsu, and Yamaguchi.
    Post war there was a drastic shortage of transportation of any kind in Japan. Something inexpensive, and fairly reliable was what fit the bill. While Honda's earliest products were bicycles fit with army surplus, two stroke generator engines, Yamaha's first production was a carbon copy of a prewar German D.K.W. 125, which was purpose built, rather than a stop gap measure. Later, Suzuki, and Kawasaki would follow with their own designs. Tohatsu was still producing the arrow in 1966, and it was offered forsale in North America. Bridgestone lasted until around 1970 and offered a range of very fast and innovative motorcycles. In their case it was lack of dealerships that sealed their fate. Hodaka was produced in Japan, but was designed in Athena Oregon by PABATCO, which stood for Pacific Basin Trading Company. The Hodaka was a well engineered, robust, and reliable bike, and ruled the 100cc class both in desert racing, and trail bike racing, which later became known as enduro. They lasted into the late 1970's. In any event, for further reading, there are very detailed histories online, and in print, for many of these manufacturer's. Below are a  pics of some of  these bikes.
Bridgestone Scrambler
  Bridgestone gtr
Bridgestone 350 gto
 Asahi Myata Circa 1954
 Rikuo Circa 1959
 Tohatsu Circa 1955
 Hosk Circa 1958
 Coleda (Suzuki) Circa 1957
 Lilac Circa 1960
 Yamaha YA-1 Circa 1955
 Meguro Circa 1959
 Yamaha YA-2 Circa 1957