Dry summer climates definitely do call for different methods. My climate has no rain from mid-May through sometimes mid-December, and I've had to customize methods. I used to use mowed weeds, but they just created too many more weeds, so I compost those now.
Wood chips are excellent for holding moisture, even when you think the climate is too dry. I don't think you can go wrong with them, unless they are from trees that have growth inhibitors, like redwood and red cedar. Even those just take longer and are better than nothing.
You might experiment with just how thick a layer works for your soil and your location. It might be more than you think, 3", 4"? Check for how moist the soil is underneath as you add them, you ought to start seeing damp soil and maybe white fungi. Then maintain this thickness as they will incorporate with the top layer of the soil, and you want to keep the soil covered completely. If there is the occasional downpour or drizzle thick woodchips will keep and hold that moisture to the soil.
If the wood chips are from very green wood it's going to take longer for the soil critters to be able to use what they have to offer, so adding nitrogen from composted manure helps, pulling the chips back and watering in about a 2" layer, then replacing the chips. Urine also has nitrogen, and can be diluted 1:1 and added to the soil under the chips, but don't get urine on plants or annual plant roots. Out around the drip line of trees, mature perennials, grapes, berries, etc. is okay.
As far as green manure, I've found it easier to stick with deciduous food shrubs and trees that drop their leaves planted in food forest style. Ground covers like native vetches are great, just let some of them go until you get seeds for next year. Mow or clip them off, leaving the roots in the soil and the nitrogen nodules they have will also improve soil.
Trailing Nasturtiums (there are some that are mounding, but aren't as helpful,) are cheerful annuals that will spread and create shade. They die back and disappear, easy to save seeds from. If you have animals I wouldn't recommend burr clover, burrs can get into ears and noses and cause problems, but I've got a lot of it showing up in general. I mow it when it gets about shin height, and it grows back in late spring, before it dies back and reseeds.
Native annual grasses are also excellent. In Africa they have now found that allowing the annual native grasses to grow holds water in the soil and the roots break down to improve soil. Mow them after they have gone to seed, and leave the roots in the ground.
If it's a new batch of chips on soil that's never had wood chips, filling in between with bagged tiny chipped wood mulch from the nursery can help kickstart things. I've even gotten some mushrooms showing up from the bagged stuff (in the greenhouse, that's how I know it's not my mushrooms.) Mushrooms are always a good sign. Let them grow and complete their cycle so the spores will overwinter. They say mushrooms are more creatures than plants, so be nice to your mushrooms :-)
Hugel trenches, a couple shovel blades deep, with very dead wood, also hold the moisture and create great growing environments. I fill in around the wood with native soil and manure, composted or not, watering it all in. Then put really rotted, pithy wood that you can twist with your hands, at about the 6" level (hand depth) so it will absorb water with nutrients and the roots can penetrate that pithy wood, and take advantage of what it has absorbed. Topping it off with compost and woodchip mulch. These have worked the best for me in general, and will last for several years if woodchips are added and break down, migrating down into the soil.