Nowadays a lot of audio versions of the Rubáiyát are available. Early recordings were done on lp in 1955 by Jim Ameche, Ralph Bellamy and Raymond Massey. These are usually FitzGerald’s versions, and some are with background music.

Recent recordings (a first selection, more will follow):

The Rubaiyat of Omar Khayyam. By E.F. Thompson. Narrated by Mark Turetsky.
Audible Inc., 2013.
Time: 2 hours, 3 minutes.
From Audible, an Amazon company.

The Rubaiyat of Omar Khayyam. Translated by Edward FitzGerald. Narrated by David Calderisi.
Published by David Calderisi, 2012.
Time: about 60 minutes.

The Rubaiyat of Omar Khayyam. Translated by Edward FitzGerald. Introduced and read by Peter Greenhalgh.
English Speech and Pronunciation, 2012.
Time: 30 minutes.
From: AudioAndBooks.com.

Rubaiyat of Omar Khayyam. By Edward FitzGerald. Narrated by Robert Bethune.
Freshwater Seas, 2010.
Time: 34 minutes.
From Audible, an Amazon company.

Alfred Drake reads The Rubaiyat of Omar Khayyam. By Edward FitzGerald.
Saland Publishing, 2009.
Time: 22 minutes.
From Audible, an Amazon company.

The Rubaiyat of Omar Khayyam explained. By Paramhansa Yogananda. Narrated by Donald J. Waters.
Cristal Clarity Publishers, 2006.
Time: 6 hours, 27 minutes.
From Audible, an Amazon company.

The Bird of Time. Selections from The Rubaiyat of Omar Khayyam explained. Voice and instrumentation by Swami Kriyannda.
Cristal Clarity Publishers, 2006.
Time: 1 hour, 1 minute.
From Audible, an Amazon company.

Another important resource is Internet Archive, which has a collection of recordings by LibriVox, read by volunteers. Here you will also find other translations than those by FitzGerald. And of course there is YouTube with lots of footage and recordings. A selection will follow soon.