So, I was able to get this to work by doing the following in PHPMyAdmin:
1) Click the "Export" tab for the database
2) Click the "Custom" radio button
3) Go the section titled "Format-specific options" and change the dropdown for "Database system or older MySQL server to maximize output compatibility with:" from NONE to MYSQL40.
4) Scroll to the bottom and click "GO".
This worked for me and I was able to import that database to a server running MySQL 5.1. This thread about a plugin gave me the idea to use that setting, I actually don't know what MYSQL40 even means.
Based on what Otto said, I guess this wouldn't work if your database has 4 byte characters saved in it but hopefully this helps you mctenold and maybe a few others.