There is only one screw holding the main pair of front fans in the Mac Pro 3,1 - but the assembly itself must be pulled square towards you, as it's slotted... and can be a bit resistant to being removed first time [especially as it's had 10 years to set itself firmly in there].
Note: If you've a full-length graphics card in, you will have to remove it first, in order to remove the aluminium cover over the CPUs. That's magnetic & just pulls away.
You'll also find the CPU heatsinks will be clogged more than the fans you're trying to clean.
This is the screw you already found...
Once that's out, pull square from the top right & bottom left - here it looks like they're using a PCI slot cover to assist, though I've never used anything other than bare hands
This is what you're trying to overcome - the drag on this channelised slot which keeps it all square, plus the fan plug/socket behind the assembly.
Picture Source: Apple - Mac Pro 3,1 2008 Service Source
The rest - experience ;)
I just looked at iFixit's take apart - though they lump the 1,1, 2,1 & 3,1 all in the same category - they mention a second screw at the bottom of the assembly; which I don't recall ever seeing on a 3,1.
However, it looks like one of those screws that the first engineer to disassemble it takes out & never replaces because it's so awkward to get to ;)
I don't have a 3,1 I can open up right now to check - & it is most definitely not mentioned in the specific 3,1 service source manual.