Android in general will only notify you with an LED notification and an icon in the notification bar. You could use a third party messaging application and try and force pop ups to show but to me that was annoying. If you choose this you can try Handcent or even possibly Chomp SMS and Go SMS.
Another option is someone made a third party application that mimics the new Motorola Active Notifications. Its called Active Notifications on the Play Store but I don't recommend this because while it does what you're looking for it also drains a lot of battery since the Note 3 isn't designed for it. Motorola has their phone only turn on a portion of the screen where as this app turns on the whole screen. Shame because it is far better than standard notifications.
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