Since this is all hypothetical, lets assume movies can tell us something about killing Zeds. The show "Walking Dead" presupposes that the only part of a Zombie's brain that is working is the brain stem, and so that must be destroyed to stop a zombie. However, the brain stem is only
1-1.5" wide, and 1.5"-2" tall -- roughly the size of a golfball. See image:

...and yet, I have never seen a head shot didn't drop a zombie, which would be far from the case if Walking Dead had it right. Based on just about every other Zombie movie and show, I think we can deduce that shooting a zombie in the brain would be similar to a live person. In the vast majority of cases, it will kill...with a few rare exceptions. With that being said, I wouldn't have a problem using a .22LR on zombies. If 95% of the time any head shot will do -- the .22LR is a pretty good contender:
- Cheap: Can stock piles shit tons of .22LR for next to nothing
- Small: Can carry shit tons on you easily
- Ultra quiet suppressed (using subsonic): As in "all-you-can-hear-is-the-clack-clack-of-the-action" quiet.
- No recoil: Miss your first shot? First head shot didn't work? Multiple Zeds headed your way? What round can offer faster follow ups than .22LR...
- Accurate (especially with subsonic, believe it or not): See above
- Brain Jumbling: If what they say is true, the .22LR MAY serve as a great round for Zeds if only because of the "bouncing around inside the head" effect
Sure, there are more powerful rounds...but we aren't hunting bears, nor are we trying to take zombies down with hits to center of mass. There are definitely drawbacks to .22LR, but what is effective against the living may not be the best choice for the undead. Hydrostatic shock, "knock down power", bleeding out -- all of these things that make projectiles affective against the living doesn't mean jack to Zombies, and so we need to change the way we think about the calibers we hold near-and-dear to our hearts. Its all about head shots, nothing but head shots...on slow moving, dim-witted, and uncoordinated "lemmings". Given this, .22LR seems like a fine choice to me.