iPad's Mail won't check for new messages in folders

I have a number of folders and filtering rules on my various e-mail accounts (Gmail, personal email, company email, etc). The problem is that the Mail app only checks for e-mail on the "received messages" folder.

That folder is where unfiltered messages end up on, and the app does check that. The other structured folders, where the filtered messages go, are only checked when I enter them manually.

Is there any way to automate the checking of e-mails for all the folders on your e-mail accounts?

Short of getting a different mail client on your iPad I can get you more than half of the way there.

Set up gmail as Microsoft Exchange server (leave the domain blank, add user and password and let it try to connect. When it fails, use m.google.com for the server and you can choose mail, calendar, address data to use ActiveSync)

Once the first mail sync starts, you can then choose days to sync and which subfolders will have push enabled. The mail badge count still only reflects unread messages in the Inbox, but you will get the mail notification sound/vibration for any new push message arriving in your selected subfolders. Your iOS device will be ready to go when you tap into the folders that have new push messages.