Hi Chris,
There are a few reasons you would get the error about constraints not needed, including selecting your IK handlers upon export -but you mentioned your skeleton is not that complicated. The other reason may be that one or more bones are not skinned to the skeleton.
So I would unbind and rebind the mesh to the skeleton. Then in the Outliner make sure you have the mesh and all joints selected -but no IK handlers, etc.- and export selected FBX.
When importing if you are still encountering errors try importing with “no skeleton” selected. It sounds strange, but I have seen this fix a couple of people’s issues as it imports their skeleton regardless.
Also, if you are using Maya 2014 or earlier, I’d be happy to take a look at your file if you want to post it in a zip file.
Thanks