What is the recommended way to backup a MySQL/Amazon RDS database to S3?
The recommended way to back up RDS is with automatic backups and DB snapshots.  DB snapshots are basically the same as EBS snapshots, which are stored in S3 behind the scenes, but are only available within the same region.  

If you need cross-region fault tolerance (good plan!), there is no way to restore your data  in another region without doing it "the hard way" from a mysqldump.  Your alternatives are to back up using mysqldump (slow and terrible for any reasonable sized dataset), or set up your own EC2-based slave in another region and back that up using any available method (xtrabackup, EBS snapshots, etc).  However, then you are back to managing your own MySQL instances, so you might as well abandon RDS entirely.

For my money, RDS provides absolutely no benefits in really any way and a whole lot of disadvantages in performance, flexibility, and reliability.  I would ask yourself what value RDS provides to you.