“Edelweiss”!
Oh, wait…Australia!
Never mind.
Yeah, this would have worked better if I’d thought of it before I posted my first one in this thread, but there ya go…
In addition to AC/DC and didgeridoo music a guy named Frankie Davidson comes to mind. For all I know he’s the Aussie equivalent of of Weird Al Yankovic but my folks brought back one of his albums (“A Real Fair Dinkum Aussie”) and I found it hilarious.
http://www.normierowe.com/frankie_davidson.htm
I recall “The Aussie Barbeque Song” and an ode to the great outdoors called “Fifty Million Blowflies Can’t Be Wrong”.
How about “Weather With You” by Crowded House? Neil Finn may be a New Zealander, but the band was formed in Melbourne. I’m not sure exactly what instruments are used in the song, but they sound Australian-esque to me.
“Please Mister Please” by Olivia Newton John?
I’d have to go with the Hoodoo Gurus.
Yeah, I know they are an English guy and an American, but still…
The stupid Outback (steak house) commercials.
I just got this album after spending a lot of time looking for good songs.
Snake Gully - Home Among The Gumtrees
Hunters and Collectors - they’d fall into the ‘bands that happen to be from Austraila’, but Throw Your Arms Around Me is sung in every pub and by every Aussie bloke with a guitar. Hell, I can play it and I’m a Kiwi sheila.
No one remembers the Little River Band?
That and Muriel’s Wedding too.
So, besides ABBA, for me, it’d be Midnight Oil & Yothu Yindi.
Oh, and the Orange Organics.
Please don’t use any Kiwi songs! The two countries are not interchangable, and you’ll end up offending one party or the other… 
These are recommended either as being really familar to Aussies, a strong accent, or mentioning parts of Australia:
John Butler Trio - Better Man or Funky Tonight
The Waifs - Lighthouse
Ben Lee - Catch My Disease, Love Me Like The World Is Ending
The Veronicas - Revolution
Josh Pike - Memories and Dust
Icehouse - Electric Blue
Koolism - Koolism Is
Colin Hay - Melbourne Song
Paul Kelly - Sydney from a 747
Warumpi Band - Stompin Ground
James Reyne - Reckless
UpsideDownUnder - 1200 Techniques
Operator Please - Just A Song About Ping Pong
John Farnham - You’re The Voice
Jimmy Barnes - Working Class Man
Diesel - Tip Of My Tounge
Skyhooks - Balwyn Calling
Yothu Yindi - Treaty or Djapana
Anything by Paul Kelly and the Messengers.
The theme song to Steve Irwin’s The Croc Files.
I was going to suggest the John Butler Trio but Girl From Mars beat me to it, so I guess I second them.
Thirding the Waifs.
Also remembered I’ve Been Everywhere, by Lucky Star (words by Geoff Mack). Later this was adapted for NZ and US, but it might be cute.
Ha! They’re about as Australian as AC-DC and the Bee Gees!
I was going to suggest the Coke song, but this would be it! I first saw the ad when I was on holiday in Australia, and loved it. The song (and the ad) really takes me back to that trip.
“Friday On My Mind” by The Easybeats.
“The Band Played Waltzing Matilda,” by Eric Bogle.
“The Ballad of Ned Kelly,” by Fotheringay.
Only the last is sung by a native Australian (Trevor Lucas), who was also the only Australian in the band.