You can wash microfiber towels in the washing machine, but make sure you only dry on the lowest temperature / heat setting and never use fabric softener. High drying temps and fabric softener (despite the name) will both make the ends of your microfibers harder, not softer. So the next time you use the towel, it will likely induce scratches and swirls in your paint.
I also recommend a detergent specifically designed for microfiber towels such as this one that I personally use:
Rags to Riches is the next generation of microfiber detergents using new breakthroughs in cleaning technology. The advanced formula deep cleans and restores cleaning and detailing products. Absolutely safe for all microfiber types and concentrated for commercial use. Use Rags to Riches to...
psdetailproducts.com
My personal favorite drying towel is The Gauntlet sold by the Rag Company:
The perfect choice for drying your vehicle quickly and safely drawing moisture away from the surface into the patented twist/plush fibers of the towel. absolutely one of the most absorbent towels by weight that we have ever tested (10x+). 70/30 blend, 900gsm, and "ButterSoft" Suede Microfiber...
theragcompany.com