Any idea what is causing these brown spots?
Not familiar with Zoysia, but know it suffers from a few fungi as does St. Augustine. We're moving into the active time of the year for some types when damage really starts to show. The irregular shape fits, but the healthy green islands don't and few, if any, blades appear chlorotic - hard to tell from the photo. Most likely something else, but would at least do a bit of research to rule out a fungus because some of them can be a pain to eradicate and total a big chunk of lawn in a disappointingly short time if left untreated.
You may get more response over on the GW Lawn Care forum. Best of luck with it.
Here is a link that might be useful: GW Lawn Care Forum
Thank you for your response. I cross posted in the forum you suggested and was told it was likely lack of water. I did the "screwdriver test" and it appears that is the problem. I've been watering more and the areas are greening up.
Glad to hear it sounds like an easy problem to fix. Looks like an attractive grass, but don't see much (any?) in our DFW neighborhood. However, the TAMU extension folks put a Palisades zoysiagrass lawn in this year at a WaterSense demo home just a few miles away, so maybe we'll be seeing more of it.