Nothing automatic. You are going to be animating position and possibly distorting a bit with an X scale animation. You will need to adjust the speed of the animation to get the look you want. When you have the first one worked up correctly then save the animation as an animation preset. Why that animation preset to a series of nulls, one for each layer. Then adjust timing a little bit by sliding the nulls down the timeline and then parent one null at a time do your other layers.
You're talking about animating the positions of various layers. They probably won't all be in the same place on the screen -- if they were, you'd only be able to see the top layer clearly!
Since there's no "one size fits all" solution for animating things in various positions, you'll have to do this: get just ONE layer blowing off the screen the way you want. Then take a snapshot of the comp window, copy the keyframes, and paste them into the next layer.
OOPS! the layer moved! No problem -- using the snapshot as a guide, select ALL the keyframes, go to the first keyframe, and use the up-down-left-right cursor keys to move the entire motion path until it matches the snapshot. You'll probably need to tweak the position of the LAST keyframe, and its position along the timeline.
Repeat as necessary for the remainder of the layers. And know that Motion Blur is your friend.
When Dave said "OOPS! the layer moved!" he is referring to what happens when you copy and paste keyframes or apply an animation preset. That is why I suggested using a bunch of nulls, one for each object you want to blow off the screen. Once you get the movement down you can apply that motion to a null, and then put your objects where ever they need to be to start and parent them to a null. You can then adjust the timing of the keyframes for the nulls to get the objects to convincingly blow off the screen.
I'd also throw in some squeeze and stretch to help sell the effect.
Thanks to you both for the replies. It seems like this will work for me. I really appreciate the tips.
One more question: I have a lot of shapes to import from Illustrator for my animation. Is there an easy way to do this in AE? I tried a suggestion I found which said to save each shape on its own layer in an AI file. This worked, however, the shapes are now rasterized so I can't manipulate the color as easily. Is there a way to get the shapes and their editable paths in tact imported from AI into AE? I'm working in CS5 btw. :-/