I'd recommend a PSU tester ($15 from newegg) before sticking paperclips in sockets, but that's just me lol
I don't know that EZ-Cool brand, but it's like 50/50 if they just turn out junk that doesn't work or if you've got a more serious problem.
If you have access to a known good PSU you can try swapping that in to test, also is everything else wired properly? (ie. Case Switch)
One thing you can try is shorting the 2 pins for the case switch and see if the fan's start. If they do- bad switch on the case.
The paperclip technique is a time tested method of checking a power supply (as well as using one to power something without a motherboard). It's funny that you should recommend shorting the power pins on the motherboard, since it's essentially the exact same principle.
And to the OP, just because stuff is new doesn't mean it works. It's not uncommon at all for computer parts to be "DOA" (dead on arrival). In this particular instance, the motherboard, CPU, or power supply could be the culprit. The power switch is definitely a possibility as well, but somewhat less common.
To test the power supply, the
paper clip method isn't a bad place to start. If you do this and the power supply's internal fan does not spin up, then it's surely the power supply that is faulty. However, if it
does spin up it still does not necessarily rule out the power supply as the problem. The very best way to test any component, as raw mentioned, is to switch it out with a known working one.
Also, as to what raw was saying about shorting the pins, what you want to do is locate the two pins that the case power switch plugs into and simply jump them with anything metal (screwdriver works fine). This simulates pushing the power button, and if the computer powers up fine then it would indicate that the power switch is defective. However, before even doing this you might want to check that the power button is even plugged into the correct pins, as this is a common rookie mistake.
If neither of these two tests show any definitive results, then the only other option is to start switching out components until you find the problem. It's clear that the power supply is the weak link here in terms of reliability, so that's really the place to start.