Hi Rick,
In 2008/IIS7+ the ApplicationPoolIdentity accounts are hidden accounts that have dynamically assigned SID's (created and assigned when the ApplicationPool is started). But the accounts live as (hidden) users under the IIS_IUSRS group on the local machine.
Give Read/Write permissions for the IIS_IUSRS group to the folder (permissions will inherit down to all folders).
If that doesn't work for you then the easiest thing to do would be to change the identity on the app pool to NETWORK SERVICE and then give that user the permissions.
Hope that helps,
Joe