Not sure I would be willing to use it on a site that has protected information on it (health care, ecommerce, etc.). Other SSL vendors put monetary guarantees on their certificates. For example, GoDaddy puts a $1 million liability protection on their certificates. I don't see anything like that with Let's Encrypt (but I may be missing it... haven't looked too deeply.)
That said, I would certainly use it in production for run-of-the-mill sites.