Mighty Hoax 5-17 is the place to grind this. You start with 7 Yellow ''Sniper'' Birds, and I calculated if you're fast enough, you will sling 26 in a minute, so it should take less than an hour. With few breaks here and then it should still be around 1 hour since I calculated 26 birds x 60 min are way more than 1000 birds, it's actually 1500.
Just make sure to send them backwards if you don't like the screen to go far away from the level (become bigger). If you send them forward the screen will follow and you will use more time because you will have to go all the way back to the next bird.
But I suggest going for 3 stars for the cheevos first, then you should have just a few 100's left so, I guess the grinding would be like 20 min or so, depends how good/bad you are getting 3 stars and how many birds you have slinged to beat the levels. If you want a quick guide for 3 stars I would recommend angrybirdsnest.com, they have all episodes. So yeah, hope this helps, and good luck