A/C motors in many parts of the world commonly rotate at 3600 RPM because that's 60 Hz. 7200 rpm is obviously twice that, and 5400 rpm 1.5x.
I don't know the real reason, since HDD motors aren't driven from A/C, but it's likely IMHO that it's related to that. Like used to be possible with vinyl record decks, it's easy to check that something is running at the right speed by illuminating it with a strobe running at the required speed. If it's at the right speed (or a simple multiple thereof) then a mark on the motor will appear stationary.
Like many things in computing, the reason is historical. The design of early PC hard drives was based on earlier, large, mainframe hard drives which were powered by AC.
That is according to PCGuide.
Correction: HD spin motors are driven by AC, very likely three-phase, but the AC comes from an inverter, although it's not commonly called that. It converts the DC power fed to the drive assembly into AC for the spin motor. That's done by an IC on the drive's circuit board.
Brushless DC motors could be used, but for mass-produced specific-purpose devices, that would'nt make sense. (DC motors with brushes are out of the question. No way to make the brushes and commutators last a long time.)
When you get up to 7,200 RPM, air resistance (and, probably turbulence) start to become significant; at 15,000 rpm, platters need to be smaller, afaik primarily because of air drag.
Some recent drives (2012, late 2011, perhaps slightly earlier) run at variable speeds, apparently dependent upon what they are doing; likely that the slowest is for standby.