What barriers prevent plastic from becoming a fiber optic, data transmission medium?
In fact, I've never even heard of such research making headlines on
  the more common tech news sites.


Tasks like that are called deveopment, not research, normally. 
And no industrial company will talk much about such work 
if its not sucessful. 
In fact there are such organic fibers for minor purposes (eg the Laserjets 
II and III used such a fiber to send a synchronising signal from the 
laser scanning unit down to the control board, for about 10 inches.
30 years ago one could buy such fibres for experiments at radio shack) 
If You read some literature on optical fibres, You will learn that 
low scattering is the grail. And plastics scatter a lot, for 
very basic reasons of polymer structure. Try to look throug a plate of 
acrylic glass "edge on", and You will understand.