How do I request a song for the Rock Band Network on Xbox 360?
Often, the best way is to reach out to the band! Many musicians and labels don't realize that they can release music through Rock Band, so if you reach out to the band and educate them courteously, you may open the door to get their music into the game! There's a helpful post from HMXHenry here on the RockBand.com forums with some suggested etiquette for contacting bands.
Another option is to post in the main Rock Band Network forum here. There are many companies and individuals (collectively called "authoring groups") offering Rock Band Network authoring as a service to musicians and labels, and if you draw their attention to a song that would be great in Rock Band, they may reach out to the artist. Please keep in mind that bands on major labels are more likely to appear as regular DLC than in the Rock Band Network.