So there is the ARM firmware and the USB host software. For Linux they provide a Cypress wrapper for libusb and some examples, that was straightforward.
For the ARM firmware, there is a short document called "Using the FX3 SDK on Linux Platforms"- it was pretty straightforward. It's arm gcc, plus a qt-based program called "cyusb_linux" to interact with the FX3's bootloader (you program it over USB).
You can use Eclipse if you want (they have some kind of support for it), but I just used the Makefile in the example projects.
The ARM firmware uses ThreadX RTOS, but the actual application is pretty small (and the space is very limited as I recall).
The only stupid thing is that they have this thing called GPIF-II designer, which I think is a Windows-only tool. The final output is a table in a C header file. I think it's a .NET program, so I think it would be easy for them to support it in Linux, but they don't.