My husband and I moved to California from Australia in January 2008 on an E3 visa working for a US company. This is a temporary visa which cannot lead to a green card. In the US, I believe we should be resident alien for 2008 but I am confused about being a dual-status resident. In Australia, should we be tax residents until December 07 or continue through to June 08? Is there any benefit?
What should our tax status be in both countries?

If you moved to the U.S. in Jan 2008, it is too early to tell you if you will file resident return or nonresident return or dual status return.

E3 is work visa and it is also subject to Social Security and Medicare taxes. Also in the U.S. you will pay taxes on all the income earned in the U.S.
If you have to pay tax on the U.S. income in Australia, you should get credit for tax paid in the U.S.

If you spent even ONE day as a non-resident alien in 2008, you and your husband will have to file as dual-status aliens.

If you spend every day in 2008 in the U.S. under the "E" visa, then you can file as resident aliens for 2008.