Whilst the monarchy does not have much of a role in day to day politics, it is a good safeguard against having a leader with too much power. Think about it, if we were a republic at the moment there would be no one to stop Howard from going crazy and creating all kinds of horrible laws. In our current system we have two people, the GG and the Queen to stop him if he went too far.
Personally I would like to keep the monarchy, basically because I love history and keeping our ties to monarchy and Europe allows some sort of connection to a grander history. Also I love royalty.
Although I might support a new governmental system with an Aussie head of state, if it was a new, fresh idea and structure that was not based on the USA system (which I fear it would be if we were to become independent).