FT2232H ground noise
You're probably seeing two things here: EMI pickup on the scope probe ground wire coupled with crosstalk in the ribbon cables.  Try a measurement with a short (5mm) ground wire - take off the scope ground lead and remove the witch's hat (retractable clip), then wrap a piece of bare wire around the exposed metal sleeve.  Measure across some bypass cap on one of the boards and see what you get.  Also check the signal lines.  You should never see 1 Vp-p variation on ground planes that are electrically connected unless you're moving around many, many amps.  

Crosstalk in the ribbon cables is a bit more complicated to deal with.  Since turning down the transmit power on the FTDI chip improves the throughput, this seems like a likely explanantion.  The best solution is to wire up the cable ground-signal-ground or ground-signal-power-signal-ground.  It may be possible to get an old IDE cable with interleaved ground wires and use that.  The 60 MHz clock line is going to be the most sensitive - it may be worth trying to send that separately with a twisted pair of signal and ground return.  A better solution would be to get a board made (or find an existing board) that integrates the FIFO chip and FPGA on the same board.