Cairns has the Smithfield XC trails and new DH course, just a short ride from Cairns itself.
Very good trails, they have been built for the XC World Cup this year.
There is also Atherton a short drive away and they are great too, the most similar riding I found to British style trail centres.
Melbourne the most popular place is You Yangs, not ridden there though. Sydney, not much riding unless you get a local to show you some places. Manly Dam is the most riding spot. If you get a chance to ride in the Blue Mountains there is some good riding there and also fantastic views.
Of the 3 places, Cairns probably has the best riding around the city. Not sure what the weather is like then though, as its a tropical area.