What parts, if any, of a chicken, turkey, duck, goose, etc. are not kosher to eat (other than the blood)?

None - unlike mammals - there is no problem with the sciatic nerve in fowl. 

Ironically, I answer this question as someone who hasn't eaten meat in 25 years or so.  

 

As long as the blood has been drained, all parts of a kosher bird can be eaten.

 

As for what constitutes a kosher bird, the birds that are kosher are listed in the Torah and must be slaughtered in a kosher way.  Particularly in a liberal Jewish context, there is also the consideration of ethical kashrut, such as determining both how the animal is treated and how the workers are treated.

 

Poultry is considered meat in the context of kashrut, and traditionally cannot be eaten with dairy.