File >> Scripts >> Export layers to files...
Here are some related questions...
Exporting individual layers in Photoshop, keeping their sizes
Export Layers to Files exports only 4 png files from 100 layers
I spent some time and wrote my own script file to automate this process.
This process is much faster than the built-in script mentioned above.
The script can be found here. (last updated May 25, 2013).
Alternatively you may view the source here.
Special thanks to escalation746 for doing some clean-up work on my script (it was really bad).
I ran this script on a 100 layer, 450MB file in under 60 seconds. Running the built-in script on the same file takes me about 30 minutes.
Please let me know if you run into any issues with the script.
Instructions for use:
File >> Scripts >> Browse (then find the script)
Here are screenshots to help if you need additional help.
IMPORTANT: The images will automatically be saved in whatever directory your multi-layer file is contained in! Feel free to make any adjustments to the script as needed.
Disclaimer: Please use script at your own risk