any organic matter you have handy will do. There will always be a debate on which wood (the essence, the age, the size, etc.) is best but really use what you have close to you.
Understanding the principle behind the huggel bed might help to make those decisions.
As always, observe, experiment and keep learning
Edit : you might want to have a look at Geoff Lawton's technique in desert environment. Huggelmound might not be the best solution for your climate considering that if it is too raised it might dry off quickly and delay the rotting/digestion process. It really depends on your situation (if you have strong winds, shade or no shade, etc..). As an immediate reaction to your “high desert“ location, I would say rather use that organic matter you have as heavy mulch and try to develop shade, water collection and build up life within your soil
sagebrush & other woody materials make excellent mulch. You do not need to chop it fine. The branches can be laid around the plants in short cuttings less than 12 inches. You do not need to measure. lol. Also any cuttings from other plants & weeds (before it forms the flowers) & leaves can be laid as mulch. Use whatever you can get a hold of.
You do want to want your wood chips to be very fine as this will seal the top layer of your mulch & your rain will run away instead of soaking in. Ex
Sep Hozier who "invented" hugel culture does lots of work too in dry lands of Spain & Portugual, so you just have to understand the difference of hugel culture in dry lands compared to wet lands and adapt your techniques.
If you land has some areas where more water washes over it because of the slopes then that might be a good location for hugel culture because it gets more water than just the rain water on that given surface. You could use swales to divert some of the rain water to your hugel culture.
If you do not have much access to woody material, then look for other organic material that will work. You only do hugel culture if you have the resources as there are other permaculture techniques that may make sense for you in your climate.
Plus, plant the type of plants that will thrive in a desert, not the the water hungry plants that are high maintenance for your climate.
Brad Lancaster books & resources (videos, podcasts and website are good resources in general for growing in desert climates as he explains lots about water harvesting in desert climates.
Thank you for the replies. I'm looking forward to using the sagebrush for my hugles now. I have several raised news that I will dig out and rebuild with the woody sagebrush in the bottom. I am also going to build hugles as berms on my property lines, so I can add privacy, garden space and shade trees. Hopefully they will hold enough water from the winter spring and fall to get through the tortuous hot summers.