Having recently deployed Exchange 2013, we discovered that you can no longer give a security group access to a mailbox using the Gui (Exchange Admin Center). As you can see this Test group is not available for selection in the Exchange Admin Center:
You can select a mail-enabled security group, but a standard security group requires powershell.
1. Open Exchange Management Shell on your Exchange Server or workstation
2. Run the following command:
Add-MailboxPermission -Identity “sharedmailboxname” -User “domain\group” -AccessRights Fullaccess -InheritanceType all
Get-MailboxPermission -Identity “MailboxID”