Do not play previews of songs in Xbox Music

Unfortunately, this is not possible. In fact, it isn't possible in the Zune software either. If I open an album that contains tracks which are only available when you purchase the whole album, it will still play the 30 second preview if you add the album to your "Now playing" playlist (in the screenshot, note the playtime at the bottom for the selected song):

Linkin Park - A Thousand Suns album only track preview

You have two options of getting around this limitation, though neither are ideal.

  1. When playing an album, instead of simply pressing the "Play Album" button, add each song which is fully available to your "Now playing" playlist, or a new playlist. Then play that playlist.
  2. Use the Smart DJ feature. From my testing, playlists created by the Smart DJ functionality do not contain any 30 second preview clips.

If you are able to play full-length content from that same album, most likely it is an issue with digital streaming rights for Xbox Music Pass. Note that on the Xbox Music Pass landing page, the fine print reads:

Compatible devices and internet required; ISP fees apply. On Xbox 360, Xbox Music Pass is streaming only and also requires an Xbox LIVE Gold membership (both sold separately). Some Xbox Music content may not be available via Xbox Music Pass, and may vary over time and by region. See xbox.com/music. For Xbox Music System Requirements and Terms of Use, see http://www.xbox.com/en-US/live/features-and-apps

(Emphasis added)

I haven't been able to find anything specific in terms of the partnerships Microsoft has with labels, but it does mention that some songs are purchase-only.