Shuffling Songs from Multiple Playlists on iPhone
Since I got the Tumbleweed badge for this, I'll respond with what I did.


Cleared all the star ratings from my entire music library.
In each of my "genre" playlists, assigned every song a rating of 5-stars
Created a smart playlist that includes every song with 5 stars. This playlist will sync over iCloud to my iDevices for listening.
Created a second smart playlist that lists every song in my "genre" playlists that doesn't have 5 stars. This playlist is for maintenance and is only available on my Mac.


This works as long as I do the following:


Continue adding songs that I like to my "genre" playlists.
Every once in a while, check the second smart playlist, and give a 5-star rating to any songs in it. It will then be empty, and the songs will automatically be added to my listening playlist.