I have an MP3 album which plays in my iTunes library and from my iPods through iTunes. But when I play it on either a newish ipod classic or an older ipod it just cyles through the album without playing anything.
Any idea what the problem would be?

Fascinating.  Perhaps the album needs to be re-ripped?  Or it needs o have the tagging cleared and then you re-tag the files with iTunes.
Not sure how to elimante the tagging so you can set it straight.  I used a tagging program in the past, Media Tagger(?), then later used Foobar2k for tagging, and they seem to be tagging MP3 files in different manners, even iTunes is different, and they do not always produce MP3 tags that are readable anywhere, except for iTunes.  There's probably some options I need to mess with to fix that, but I'm too lazy lately and just stick with iTunes for tagging.  =p
Or something else is the issue with your album.