Currencies cannot be compared in absolute terms, so it is not accurate to say that the Australian dollar is "worth more" than the US dollar just by saying that the conversion rate is 1 USD = 1.27 AUD. The question is, what can you buy with 1 USD. IF you can buy more in Australia with 1 USD (i.e. 1.27 AUD) than you could buy in the US for 1 USD, then the USD is worth more than the Australian dollar. If on the other hand, you can buy less, then the AUD is worth more. It's all a matter of what you can get with your currency - not how many of another country's currency you can buy - because their prices may be adjusted accordingly. To use a better example, consider the Baht. 1 USD is equivalent to 33 Baht, but it is certainly not true that you can buy 33 times less in Thailand with 1 USD than you could buy with 1 Baht (in fact, it's the opposite, meaning that you can usually buy 2 to 3 times as much with 1 USD in Thailand than you could buy in the US, so the USD is strong against the Baht). A lot of people's confusion is driven by the fact that currencies are sometimes named the same (Dollars, for example) and often times the exchange rate is close to 1 to 1, which makes people think that 1 to 1 would be parity (even though it may not be), while 1.2 to 1 means that one is "worth" more than the other (which it doesn't necessarily).
So in short, you should check out the real estate market in Australia, check consumer prices for common staples, and determine whether it means that you can live better there on $350,000 USD than you could live in the US. Of course, people that travel back and forth could probably tell you where their mooney goes farther - hopefully one of them is watching this board.
In addition, one other thing you need to consider is how you will continue to make money. $350,000 may be enough to buy a house, but you still have to pay the taxes and utilities and insurance and food and gas and car maintenance and all that other stuff. Are your skills marketable in Australia? What can you make compared to what you could make in the US? If you've never been there (which I assume you haven't if you are asking the question), are you sure you want to jump right into buying a home without checking whether you like it or not first. I would take a chuck of the money and go there for a few weeks, then see if you are ready to plunk down the money on a house.