I have a requirement to allow a user to record an audio file using their microphone, but it has to work without flash as it needs to work on iOS (mobile safari), Android browser or Chrome, and a modern browser on a PC/Mac.
Is there a clean, simple HTML5 method for recording audio and posting to a server? I haven't been able to find anything.
You can use the HTML5 WebAudio API.
An introduction to audio and video capturing Capture audio & video in HTML5
A good library to record audio with samples Recorder.js
A complete and working sample using Recorder.js How to record audio in Chrome with native HTML5 APIs
Other WebAudio API demos HTML5 Web Audio API Demos and Libraries
Supported browsers Can I use Web Audio API?
Regarding to send the data to other server, using Recorder.js you can get the audio buffer, then you could use XMLHttpRequest to POST the arraybuffer or blob to the destination server directly or encoded as base64.
MDN: Sending and Receiving Binary Data
Html5Rocks: New trick ins XMLHttpRequest2, sending data