I had this exact issue last night! My HDD was/is dying so I built a new server from scratch (my old one was about 5 years old). I reinstalled a clean FreeBSD 10.3 and then had to install all my other software. I’d backed up most of my conf files so getting Apache/Samba/Unbound/ISC-DHCP/etc working wasn’t a big chore. I hadn’t backed up my letsencrypt directory, though.
During a brief moment the old HDD was functioning, I used rsync to copy the entire letsencrypt directory to my new machine. I used the “a” (archive) flag, which also covers the “l” (preserve symbolic links) flag. From the old machine as root, I pushed the directory to my new server as a standard user.
Once transferred, I then recursively chowned the directory back to root, and I that was that! I know I could have just started again, but I enjoyed keeping my original setup and Let’s Encrypt account data