"But," says Dunham, "if you listen to the whole act you realise he's not making fun of Muslims. He's not making fun of anybody. He's making fun of terrorists, because this guy's an idiot.
"The way I imagined him was, of all these [terrorists] who are doing these things, there has got to be one or two of them who are saying, 'Should we really be doing this? I like cooking. I'd rather be a cook'. That's where Achmed came from.
"And I think I get away with a little bit more because it is coming through a character in the same way South Park does. They get away with unbelievable stuff because they're animated characters and they're little kids. In that same vein I can get away with more than a straight stand-up guy does because I have those characters."
Achmed, Walter and Peanut will be just some of the puppets Dunham brings to New Zealand for the first time as part of his Controlled Chaos show at Vector Arena on Sunday.
The story of how Achmed came to be goes back to a year or so after 9/11 when many people in the US, including TV show hosts like Jay Leno and David Letterman, started making jokes about Osama Bin Laden.
"Because we hadn't caught him. We didn't know where he was. And obviously that was the big goal, to get that guy. So our country was healing and trying to move forward and comedians would make jokes about Osama Bin Laden and terrorists, and I just wandered if I could do that too.
"But I wanted to take it to a place a little bit different and create the dead Osama because I knew where he was - he was actually killed, but he's undead and he's been hiding out in my suitcase with all my other dummies."
He used this Osama character for two years, then when Bin Laden started becoming a "tired old subject" he packed him away in his suitcase for good. But then, in 2007 on his Spark of Insanity special on TV channel Comedy Central, he introduced Osama Mk II who the world would come to know as Achmed.
"So instead of offending one person I thought I'd offend a whole group of folks," he jokes. "But we really don't know where Achmed is from though. We don't know what that accent is. It sounds a little bit Middle Eastern. But it's not even a real accent. I don't even know what it is."
Dunham is big, with the Youtube video of him introducing Achmed clocking up more than 200 million views, and he's relatively well-known here if performing a Vector show is anything to go by.
But it's not been easy being a ventriloquist and it took him years to break through. His obsession with puppets started as an 8-year-old when his parents gave him a Mortimer Snerd dummy (a character created by American ventriloquist Edgar Bergen). By age 10 he was performing to anyone who would watch and while at university in the early 80s he started performing in comedy clubs.
Throughout that decade he kept up a relentless tour schedule with his goal to one day get a slot on The Tonight Show with Johnny Carson.
"In the 60s, 70s, and even the 80s, if you were on The Tonight Show, just one time, and if Carson liked you and you did well, things were off and zooming."
Dunham jokes that when he "finally got there in '92" although it wasn't quite the bastion of comedy it once was, the show was a definite turning point in his career.
"It gave me a stamp of approval in the comedy world so that's where it started for me, but after that it was many more years of doing it the grass roots way - city after city, club after club, theatre after theatre. That built this very loyal following I think."
Later still it was DVDs like 2006's Arguing With Myself, Spark of Insanity which starred Achmed, and the boom of YouTube, that took him to a new level of popularity.
"It was a perfect storm of technology, it was DVD and the YouTube world mixed in together with Achmed. That's when it all took off."
As for Dunham's relationship with his puppets, he says though they do take on a life of their own and become very real when he is performing, there is no doubt in his mind that they are dummies.
"I can put them in the suitcase at night and not feel regret," he laughs.
LOWDOWN
Who: Jeff Dunham
What: American ventriloquist and comedian and his crazy cast of puppets
Where & when: Vector Arena, Sunday night
- TimeOut