The Facebook Graph API lets you grab a JSON representation of home (News Feed) & feed (Wall).
How do I just get the posts made by my Facebook app?
Afaik thats not possible with just the Graph API. But you can just use a FQL statement to retrieve the wall/feed, too. With this technique you are able to resrict it to the posts made by one actor_id (which should be your app ID in this case):
SELECT post_id, target_id, message FROM stream WHERE filter_key in (SELECT filter_key FROM stream_filter WHERE uid=me() AND type='newsfeed') AND is_hidden = 0 AND actor_id = 'MattDiPasqualesAppID'
Thing here is that it won't return a JSON, but a XML representation of the result. With phps DOM class you are easily able to convert it into a JSON format or any other representation you want to have of the result!
As you might be handling the whole thing with PHP anyway, you also could just take the json array, parse it into an array and filter the array with keys of your appID.