Edit: Maybe flashing the R9 290 with a R9 290X/390X vBIOS will work? (Is it even possible for the reference R9 290?)

Wot other "juicy"cards did u ran on that Intel board?
U can have an HW issue on the PCIe slot, by NOT delivering the correct power amount the slot.
With all the flashes u done it should had already give any sign...even a corrupted display screen.
There was a pcb mod necessary to the 280/x in order to enable PCIe 2/3.0 on the Mac Pro...(dont remenber well) have u checked any changes in the pcb circuit of the card

haha non of them were juicy, only simple cards without extra PSU connectors, only the PCIE itself. So no juicy..
I also suspect that's an hardware problem because immediately on boot the fan of the R9 290 starts spinning at a very high RPM rate.. (Or that could also indicate a BIOS problem as well?)

That fan spin at high rpm doesn't mean nothing, its common on many models and nothing to do with mboard, its pure graphics card initialization.
And i do not point for an issue related to mboard bios.
The R9 is a hungry board...whenever possible u should test any other card that requires auxiliary PCIe power to confirm my suspicious, good luck.