With S3 CloudFront Streaming and Private Streaming you need to convert mp3 audios into m4a audios (iTunes - CD format) in order to get it to work on your site. For some unfathomable reason, mp3 is not reliable for streaming on S3 CloudFront . The support staff of S3 has no conclusive answer to this for the moment, therefore it is best to download an application that can convert those audios. AAC is no problem as long as you treat it like a video.
Here are two audio converters (Mac & Windows) capable of converting mp3 to m4a for streaming and private streaming. Both have a trial version for testing. I tested them both, they are intuitive to work with and produce decent quality audio:
Audio converter for mac - $39.95 retail price
Audio converter for Windows - $23.95 retail price
Both work fast and they can convert a whole range of audio files in one go. The best way to work with streaming audio is to prepare a Streaming CloudFront distribution before you upload the audios to the bucket. For private streaming, this is very important. Otherwise, you will have to set the CloudFront Origin Access Identity to Read for every file that was uploaded before you made the distribution private.