How to Download to Computer a Voice Memo Received via MMS on my iPhone?
Forward the MMS to your E-Mail.
The Email client of your computer can then extract the attachment.

In the SMS/MMS view, touch edit in the upper right-hand corner.
Touch the circle to the left of the audio clip(s) you want to keep or send to others.
Touch forward at the bottom and then type in the email address that is linked to your phone, or touch the + and you can pick any of your contacts and send it to their cell/email.