This is possible, but you have to "teach" the accent. Do the following, based on your desired accent:
1) for a British accent: hold down "command-option" and yell "Jolly Good!" as loudly as you can
2) for an Aussie accent: hold down "command-option" and yell "A Shrimp on the Barbie", also as loudly as you can
3) for a Chinese/Asian accent: hold down "command-option" and yell "Me So Horny, Me Love You Longtime". You must yell for this command also; for some reason, it seems to work better if you open your window before yelling.
I'm genuinely interested in the Aussie accent, so I tried, at the top of my lungs...
But didn't work. The neighbours just shouted back "Mate, what are shrimps?"
I then tried: "Aww cumon mate, I'll buy you a beer and meat pie"
That also didn't work.
To answer the OP's question, unfortunately you can't change the accent with what Apple provides, unless you're able to find a 3rd party voice with the accent you're looking for.
As TheReef says above, if you want to change your system's voices, you need to get new voices from somewhere--and the ones referenced in the linked article are the only ones available that I know of. I have the British English voices pack from AssistiveWare/Acapela, and they're very good. Pretty big download, though, and pretty expensive--it's a lot of coding, I guess, for a fairly niche market. They're aimed mainly at people who need to use VoiceOver to make use of their Macs or who use a text-to-speech voice to communicate (some of the stuff these folks are doing with making iPhones and iPod touches work as assistive communication devices is amazing--much cheaper than the 'real' gear put out by the medical devices monoliths of the world). So for the more casual user, the price seems steep. Personally, I use text-to-speech to make audiobooks out of long webpages I don't particularly want to sit and read--means I can listen to them while doing housework, cross-stitch, etc., so I'm definitely not their target market.
One huge advantage of the AssistiveWare/Acapela voices is that you're given some control over pronunciation. You won't have much luck trying to get around some of the, ahem, features-not-bugs built into OS X's speech-to-text engine--you'll never be able to stop it thinking "sat." (as in "Okay, I thought, and sat.") means "Saturday". But you will have enough control to correct pronunciations that annoy you, and to teach it new words that it can't currently pronounce (eg you'll be able to teach it to say "drat" instead of "doctor-at"), to pronounce your colleagues' names correctly, and so on.