I'm pretty sure you could grow them just fine there.

You're biggest problem out there might be getting wood. You don't need a lot, but we are limited by the types of wood that work, and I'm not sure what you'll have available. I don't know, but California Oak might work. There are several types of hardwood that do work, so it may not be a problem at all.

From what I've gathered, here's what I can tell you. They'd probably fruit good outside in the winter there. Out there, you could probably do good with keeping the logs in the shade and soaking them in a tub of water for a few hours every week or two during the hot and dry months while you're waiting for the spawn run to complete. That takes between 6-12 months. You'll want to use water with no chlorine in it, so you might fill a tub and let it set in the sun for a couple days, then put it in the shade to let it cool down before you put your logs in it.

Once the spawn run is complete the logs will be ready to "fruit". At that point you can start forcing the logs to fruit every 8-10 weeks if you control the temperature and moisture content of the wood.

During the warmer months, to force them to fruit you might bring them inside your garage or kitchen for a day or two where it is cooler, and then soak the log in cool water for 24 hours. Keep them inside while they produce mushrooms for the next week or so. Then you could let it rest outside again for 8-10 weeks.

If you wanted fresh mushrooms every week you'd need 8-10 logs and you'd set them up on a schedule to force one to fruit every week.

Check out http://www.fieldforest.net for what types of wood and spawn might work best for you. They've got a lot of different types and strains of mushroom spawn available and info, and sources for info, on growing mushrooms.