For the last several years we've used divshare without problems for uploading and then accessing embed codes for MP3 files. I tried it again today (5th Feb) just to check its still working - and, for my account at least (such sites sometimes take away free options for newer members), it is, as these screenshots demonstrate.
Sign up is simple, ditto uploading - remember, its for audio files NOT movie files.
|Logged in to my divshare account, I upload, find the file, and (as with YouTube) simply click the SHARE button|
|This generates the embed code|
|You can also click on ADDITIONAL SETTINGS below the code. If the player is too wide, reduce the width="335"|
There are other options if you don't see the SHARE button on divshare. Soundcloud works along similar principles, though it also has a handy record button so you can record podcasts directly to the site. For this example, I played a YouTube clip then hit the record button...
Here I tweaked the code I pasted in, changing the frame width to 70% and the height to 83. As you can see, it still works, you just don't see part of the soundwave graphic, and the icons are changed. You can play around with the size settings to make it fit according to your needs.
I haven't tried it, but freewebspace.net is another that came up from a simple free upload embed mp3 google.