Mac Pro 3,1 no signal to monitor
Assuming that it is definitely a Mac Pro 3,1 (i.e. early-2008 Mac Pro model), the logic board should contain some diagnostic LEDs. These were used by technicians for troubleshooting purposes, so you may be able to leverage off these to isolate your problem.

The following chart explains each LED:



As you can see, normally LEDs 3, 4, 5, and 6 are off. In other words, they shouldn't be lit up unless there is an error. If any of these are lit up, then you know you have the corresponding problem explained in the Indicates column.

As for the other LEDs (1, 2, 7, 8, and 9) you will need to press the DIAG_LED button (it's next to the LEDs) to display them. Use the Color column to determine the status (i.e. this indicates the LED colour).

I would check all of them, but obviously you'll want to pay particular attention to LED 7. If LED 7 is not lit up as green when you press the DIAG_LED button, then you know the GPU hasn't been configured.

Once you've done this, let me know how you went.

[UPDATE]

Okay, since you're only getting LEDs 2, 8 and 9 light up upon pressing the DIAG_LED button, we know that the computer isn't recognising the GPU because LED 7 is not green.

One of the issues you mentioned in your question was a potentially faulty power supply. I would actually re-install the one you bought on eBay. The reason for doing this is that some GPUs need more power to function, which is available from connectors on the logic board. And if you have any question marks over the original power supply, it's probably best to put it aside (although I acknowledge the one you bought was a used one).

Now I have seen instances of the ATI Radeon HD 5770 needing to have the auxillary power connected in order to run. So it may be worth looking for the connector on the logic board (usually it's close to the front of the board, under the plastic slot holders). So I'd plug that in (if not already) and see if that gets it working for you. Make sure you check the cable is connected properly at both ends! 

If not, in terms of general troubleshooting when there is no LED 7 lit up, you would want to try each of the following (NOTE: I know you've already done some of this, but for the sake of completeness and other users I am still including them):


make sure the power supply and its cables are properly connected
make sure the graphics card is seated correctly 
try the graphics card in each of the PCI slots
try a different graphics card
try replacing the PS3 power supply cable (as this is what supplies power to your PCI slots)


Normally if you have no joy after all of the above troubleshooting steps, I'd say it's time for another logic board.