I wasted the better part of a working day on this problem before figuring it out...
When you add files to App_Code, Visual Studio sets the "Build Action" property to "Content", instead of "Compile" which it does for all other classes. So your classes don't get compiled, and then VS complains that it can't find them. Presumably this is VS' way of saying that App_Code is not where you should put your app code, reasonable as it may appear.
The solution is to right-click on each file and reset "build action" to "compile". This also has to be done whenever you add files. Or you could submit to VS and put your source somwhere else.
< /frustration>