The problem is this: 123D Design exports all the separate bodies as separate solids within the STL file, but Makerware only reads the first solid in an STL.
You can verify this by opening your STL in a text editor, and looking for a pair of lines that looks like this:
endsolid
solid
Try deleting them, and your STL will load into Makerware.
Another option is to do a Combine (not just a Group) operation in 123D Design, that will turn all the bodies in your file into one. It does limit your ability to edit, so you might want to save your work as a copy before doing the combine.
I've been asking Makerbot to support multiple solids within an STL for a while, it would be a good way to support multicolor and multimaterial printing.
Please get on their case about it too!