Yes, there are special statuses for spouses and children (dependents) of E-3 visa holders which allows them to travel with you on an E-3 (whether they are Australian citizens or not).
For spouses (must be married – de facto doesn't count in the US), you automatically receive working rights on the E-3S status. For children (unmarried and under 21) there is a special E-3Y status which allows them to study, attend school, but not work. More information about these visas can be found here.
Unfortunately these are the only categories available to family members, so you can't bring other family members with you.