There are not that many places to eat outside of Miyanoura and Anbo as most tourists eat at their accommodation so be aware of this if you find yourself in the south of the island.

Here are a selection of eating places and of course there are lots more in the book.  

Il MARE イルマーレ 
An authentic Italian restaurant serving pizza and pasta close to the airport.

Location: KOSEDA (See Google maps)

Freshly made muffins, soup, coffee and cake near Anbo port. Click the link to see the menu. 

Location: ANBO (See Google maps)

A great place to try fresh local dishes like flying fish and noodles. Near Anbo port.
Location: ANBO (See Google maps)

A small roadside cafe that offers delicious set lunches and coffee.  
Location: KUSUGAWA (See Google maps)

We love this simple bar/restaurant for the eccentric owners. They serve flying fish burgers, noodles and large beer.  
Location: ANBO (See Google maps)

Awaho restaurant, YakushimaAWAHO あわほ
This new restaurant is quite unique. During the day you can buy the bento meals packaged on display and eat them directly at the tables. You can also order from their simple cheap menu. Right in the center of Anbo. 
Location: ANBO (See Google maps)

For lots more eating places and more information about Yakushima - check out the one and only Yakumonkey Guide to Yakushima