Well - the way that we did it was to book a space in a campervan park for a week - most campervan parks have cabins as well as spaces for vans and tents, and they are a lot cheaper than a motel. Our one was $70 / night, and we had a discount card which took it to $63. It was very basic - a room with a double bed in it, and a kitchenette area with a stove, fridge, kettle, microwave, table and chairs. They did also provide cooking equipment, so we didn't have to rush out to find plates and cutlery - a good thing when you've just got off a 27 hour flight!
Depending on where you are aiming for, you may have to use public transport to get you to look round rental places, but at least you are on the ground to look round rental places (which is incredibly important - you need to be able to see things like which way the house is facing for the sunlight - most places here aren't double glazed and insulation is minimal, so the heat in summer and winter is important! Also, we did look round a couple of places which were lovely on paper, but absolutely *vile* when we stepped inside! You are already putting yourself through the strain of moving across the world - you really don't want to add to it by having somewhere horrible to come home to after work, even if it is only for a few months while you buy a house.)
You can also check out the local papers and trade websites which will have rentals from the landlord directly as well as the agencies (we ended up renting directly off the landlady, which meant that we don't pay any agency fees)
Also note that rental places here don't always come with white goods (unless you are renting furnished) - we had to buy our fridge and washing machine - so do factor that in to the costs of renting.