They didn't. Not really. American accents are a LOT closer to the English accents of the 1600s and 1700s than contemporary British accents are now.
Irish accents? Same deal. They have more in common with Elizebethan English than anything around today. Australian accents are heavily influenced by the urban SE English accents of the 19th and 20th centuries.
This is not to say that American/Irish/Australian accents didn't develop on their own, they just generally hark back to the accents of the first wave of colonists. If you don't believe me, look it up.
There's an assumption by some people that because Americans and Canadians still use the 'r' in all positions that their colonists must have predominantly come from areas of the UK that still use the 'r' in all positions. This isn't true - in fact almost all rural areas in Britain, including England still used 'r' in all positions as recently as the 50's. The loss of the r came from pressure on the urban middle class to mimick the upper class, which had begun to lose the 'r' in the late 18th century. This is heavily related to the education system at the time in Southern England. The loss of the 'r' in England (most of it, some areas still use it) is a very recent phenomenon.
The reason Americans didn't see this change* and we did it because they lacked the same social pressures. Australian colonists generally came from areas which at the time had completed the shift from rhotic to non rhotic r. The shift was haphazard and took much longer in areas like the North and West of England.
*Some areas did, such as parts of the North and South East. In the case of the North it was generally due to the same societal pressures as in England - the upper classes of the North - Boston and New York for example had started to ape those of the UK as it was then the leading superpower. This filtered down into the lower classes.
In the case of the south, the white landowners of the 19th century would often send their children to be educated at English boarding schools, which started the trend to non-rhoticity.
Ironically, I live in one of the areas of England which doesn't drop the R and my family are from somewhere hundreds of miles away which doesn't drop the R. Still in England. Vastly different accents though, although both still have examples of rhoticity.
You should check out
sounds familiar? it's a website that has some of the oldest recordings of British accents, from people born in the 1800s. They're like listening to another language.
http://en.wikipedia.org/wiki/Rhotic_and_non-rhotic_accents
The red area was rhotic in the 1950's.
And now (old people from the above image still retain their rhoticity):
I have no idea either hahaha. Mel Gibson and Chuck Norris perhaps?
