Given that names are considered protected health information (PHI), these could fall under the Security Rule requirements. You didn't mention whether or not employee names are considered PHI at your organization. If so, what I mention below will apply to both patient names and employee names. This issue would most likely fall under the "Transmission Security" standard in the final Security Rule which must be "addressed." What this basically means is that you'll have to perform a risk assessment to determine whether or not this PHI is at risk to any known information security threats or vulnerabilities.
If you're sending your e-mails in clear text you can bet it's at risk, both during transmission and once it arrives at its destination. So, given all of this, there is no rule that states you have to omit names from e-mails. However, you do need to make sure that any PHI (names, addresses, phone numbers, etc.) is not at risk when e-mailing it. If your risk assessment shows you need to protect e-mails, you'll need to either eliminate the PHI from the e-mails or somehow encrypt the e-mails so that the information is not interceptable or readable by a third party.
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This was first published in March 2003