module has no attribute

The problem is submodules are not automatically imported. You have to explicitly import the api module:

import myproject.mymodule.api
print myproject.mymodule.api.MyClass

If you really insist on api being available when importing myproject.mymodule you can put this in myproject/mymodule/__init__.py:

import myproject.mymodule.api

Then this will work as expected:

from myproject import mymodule

print mymodule.api.MyClass 

If you are an idiot, like me, then also check whether you didn't name your python file the same as the module you are trying to import.