Note: If you are having problems with the "ChatterEmailHandler" Apex Class not being available for lookup. Go to Develop => Apex Classes, click on "ChatterEmailHandler" class and copy "ChatterEmailHandler" text from Constructors => Signature section without "()". Now copy this in the new Email Service lookup field.
Note: Share this address with all your internal users, no need for creating separate addresses. Make user the "Accept Email From" is set to "myCompany.com", so only emails from your company domain are accepted.