Is there a cheap 3-wire "brushless" computer fan board?

Those are 12v fans with tach output. If you dont need the tach, Connect positive side to +12v negative side to drain of n channel logic level mosfet, source of said fet ground and gate to arduino pin via 100 ohm resistor, and 10k between gate and gnd.

If you do need tach output, you can't switch the low side safely or else the tach output will rise to 12v.... so need to switch the high end. So negative to gnd, positive to drain of p-channel fet source of that fet to +12v, gate with 10k (maybe 1k is better if pwming) to drain and to the drain of an n channel logic level signal (ie, rated for low current) mosfet; of that fet, source to ground, gate to arduino pin via 100 ohm resistor and 10k pulldown to ground. And tach output to arduino input pin, via say 4.7k resistor for safety, it might need to be input_pullup.....

Either way nothing fancy needed to drive it - at most the usual pair of fets for driving high side of something