This link will take you to 508 aqua phonic videos from all over the world. No since reinventing the wheel.
http://aquaponicscommunity.com/video
One guy in the Philippines has experimented with the floating raft system in a pond with poor results. The roots being in an unfiltered pond environment collect the solids and turn blackish and slow or stop the growth. In a side by side experiment he used the same pond filtered water to grow lettuce on an identical floating raft and the lettuce was many times bigger and robust compared to the unfiltered raft with only 50% even survived. So filtering is the key I think. Looking at the Aztecs floating grow gardens and some Asian floating raft grow gardens I have not found any that had root growing directly in the water but the grow medium was built up on the rafts. So I am guessing that some of the grow medium in these raft gardens was partly in the water allowing a capillary movement of water with nutrients upward to the root while filtering the water of solids. So my thinking is a system like the picture below only bigger with a grow medium perhaps of a bale of sphagnum moss and maybe mixed with vermiculite for water retention. A few inches of medium under water and 8 inches on top.


Below is an impressive system in India
http://aquaponicscommunity.com/video/urban-food-garden-update-dec