I have American dollars in an Australian bank account, which doesn't accrue any interest.
Is there a way of getting interest on it?
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Don't talk to us, talk to your bank, as harsh as that sounds. You aren't taking part in the interbank money market and any conditions you get for your account are up to negotiation, there's no law (in Australia, other than the law of the market) that obliges a bank to pay interest to anyone.
If you really want something that compensates you for the loss of interest buy an ETF on the fed funds, or other interest rates. Or buy, if you can, American government/municipal bonds.