In order to avoid the suffix you have to grant the respective user the "Send As" permission on the relevant object. This is different from assigning "Send on behalf" rights which is what happens when you declare a user to be the delegate of an account via the Outlook UI (you can also do this via the Delivery Options button on the Exchange General tab of the corresponding AD object's Properties window).

The steps necessary for assigning "Send As" permissions differ slightly depending on whether you want to send on behalf of a shared mailbox account or a mail-enabled Public Folder:

Shared Mailbox account (also works for other AD objects like Security Groups and Distribution Lists):
In order to get at the relevant options you will first have to enable the Advanced Features option (available via the View menu) in the AD Users and Computers MMC. Now bring up the relevant account's Properties window and switch to the Security tab. Here you can add the relevant users or groups and grant them the "Send As" privilege.
Mail-enabled Public Folders:
Using the Exchange System Manager, bring up the Properties for the public folder and activate the Permissions tab. You can now grant the "Send as" privilege via the Directory Rights button.

Note that in both cases you will often have to restart the Exchange Information Store service for the changes to become effective.

There is no harm in assigning both "Send on behalf" and "Send As" permissions on an object but the "Send As" will take precedence, i.e. when both are set then no suffix will be shown.