ISA slots are powered directly from the power connector, I never payed attention to PCI but I suspect it's the same. With a needle probe (and PCI pinout handy), check continuity between ATX power connector and corresponding pins inside the PCI slot. I'd do that first on a working board, then on the problem one. That would give some hints at least.
Are voltages ok at the power connector?

Is this the super cheap ISA/PCI post card? It seems not since you're using PCI and that can't be reversed, but I had the same problem today as well - however it was just a sily reason: I had inserted the ISA edge in the wrong direction.
Thankfully nothing blew up and the two motherboards I was testing still work.

All the voltages in both the ISA and PCI slots should be powered directly by the AT / ATX power connector, so you can check for shorts there? I would give it a go without the RAID card next.

I put the PCI POST card into another motherboard of the same chipset (Tyan S2567) and, both, the +12V and +3.3V LEDs illuminate. Note that the POST card is installed into a 3.3V PCI slot.

There are two 5A linear voltage regulators on the board: RC1585M and SPX1581T5. The 1585 outputs 1.5 V and the input reads 3.3 V. The 1581 outputs 3.2 V and the input reads 3.3 V.

Each CPU contains two corresponding n-channel MOSFETs. The voltage between the rain and source per pair read 3.12 V and 1.79 V. I currently have coppermine CPUs installed and I beleive the 1.79 V corresponds to the voltage of the coppermine.

The motherboard has two ATX power connectors, which make it easy to read the power-on voltage of the PSU at the motherboard.

12V = 13V
-12V = -13.13V
5V = 4.96V
-5V = -5.3V
3.3V = 3.28V

These numbers, along with the numbers of the linear regulators and mosfets seem normal to me. Are they?

So what is causing 3.3 V not to reach the PCI slots? Is there a on/off voltage relay controlled by the north bridge perhaps?

I checked the 3.3 V pins on the entire PCI slot and 3.3 V is reaching the slot, so there must be some other check that the POST card is performing prior to engaging the 3.3 V LED. On the Tyan board, I've noticed that the +12 V LED comes in about a half second before the 3.3 V LED. So there must be some routine that the motherboard is getting hung up with.

I'm at a loss. Any ideas to test?