Here are some things you should know about FBX export and Maya.

• Every animation has to be based on joint animation.
  (This is not entirely true, but in general you want to use joint based animation.)
• Save a new version, call it _baked or something like that.
• You can use constraints, but you have to bake its effect onto the objects or joints (Selects the objects or joints, Edit -> Keys -> Bake Simulation).
• The FBX file can’t contain multiple bind pose node. If you have more than one (probably) delete those (turn off the DAG Objects Only feature in the Outliner, select the bind pose nodes, delete) and generate one bind pose node.
  Set the character (or your other animated objects) to bind pose or the default t-pose, select the root joint than the mesh or meshes, and run this script:
  dagPose -bp -save;
• You can delete every constraint (since you baked it earlier).
• Put the whole skeleton in a group.
• Select the mesh (your character or other animated objects) and select the skeleton group, File -> Export Selection, choose FBX (2014/2015).

Some other advice:

• Use Lambert, Blinn, Phong or other not render engine specific shader.

I hope this is helpful to you.

Cheers, D