Here's a simple way to record an Mp3 file and embed it into your website. First sign up with http://www.drop.io This is a file storage and sharing website.

"A drop is a 'discrete' (yes, "ete") chunk of space you can use to store and share anything (pictures, video, audio, docs, etc) privately, without accounts, personal registration, or email addresses. Our site is not searchable and not 'networked', only people that know the URL of a drop can access it. You can create as many as you want and set things like a password, whether visitors can add to the drop, and how long you want it to exist." (source: http://drop.io/faq#2)

They give you 100MB of free space. Although there are lots of free file storage places out there, I chose drop.io because it is a Firefox addon. Once you have your drop made look at the right side bar. You should see a voicemail phone number. Call this number, record your message and it will appear in your drop as an Mp3 file. Now view this file and you will see "embed code" just copy and paste this into your website. Below is an example of the default way it looks on the page: