I domain management company is telling one of my clients that, starting 2018, Google Chrome will start warning visitors that their Let Encrypt certificate isn’t safe. According to this company my clients need to upgrade to some other cert from i.e. GeoTrust.
Nope, Let's Encrypt will continue to be trusted for many years to come. I'm not sure on what basis someone would be making that claim. If you're willing to share the name of the service provider saying this, I'd be happy to reach out and try to correct the record.
If DigiCert is still issuing new certificates under the GeoTrust brand, they're presumably using acceptable roots, so those new certs won't be distrusted. (Unless the subscriber requests otherwise.)
Not sure where you heard this but this is totally untrue. I imagine that the story of Chrome UI distrust has been spun into something else. Here’s what we know…Chrome will be marking NON HTTPS websites as insecure, Chrome is also moving towards a distrust of Symantec certificates.