We have five rental places during our big trip. Most of these rental places have required a banking deposit and not a deposit paid with a credit card.
At first I worried how to do this without incurring high transfer fees. I eventually found the website Transferwise. This site is great when you need to send someone money in a different currency. Recently I sent around $750 USD and the fee was only around $5.00. If I had paid by credit card I would have incurred a $14 fee, so the transfer by bank account has made things easy.
I get the feeling that people in Australia and New Zealand are used to bank-to-bank transactions. In the US we seem used to credit card transactions. Another thing I have noticed is that if you do paying credit card, most places assess an additional 2% fee, so it is better to do the bank-to-bank transactions instead of paying by credit card.