Look familiar? Foxtails are the bane of many a trail runner, and they're an even bigger hazard for dogs. The topic came up because when I was volunteering in Marian Bear yesterday, the veteran I was following wondered if foxtails were native; I'd always understood they were not, so I resolved to look it up. Like many grasses, the a species can be hard to identify among many for amateurs like your blogger, but it seems obvious that our California foxtails are Hordeum murinum. These look just like the ones I find in my socks and shoes and sometimes elsewhere in my house, and it's listed as a state-wide invasive on CA flora, which come in to disturbed grasslands, especially those disturbed by drought.
To answer the question, they are indeed invasives (from the Mediterranean), not that this makes them any more annoying or dangerous to animals. (Poison oak is native and somehow I don't like it one whit better because of that.) Some of the native grasses include bunchgrass (below), which is discussed here at the California Native Plant Society; you can often see it planted at public buildings in California (there's some around UCSD) and also in my experience, at higher altitudes where the invasive European grasses can't survive, above nine or ten thousand feet. (If you really want to know about California native grasses, here's a whole book on it!)