After reading everything I could find by Carlos Ruiz Zafon, and a few other novels translated from Spanish into English, I have come to the conclusion that modern Spanish literature is something I would like to find more of. Especially the ones set during and just before and after the Spanish Civil War.

Any suggestions? Obviously the books would have to be available in English translation.

Though he is from Portugal rather than Spain, you might enjoy Jose Saramago's writing. I particularly liked 'Death With Interruptions' and 'The Stone Raft'.

The Time of the Doves, by Merce Rodoreda, is an excellent novel that has been translated into English.

I have one on my bookshelf.  The Gaudi Key by Esteban Martin.  Takes place in Barcelona in early 1900's and present.  Translated. 

A couple others from Spain, set in the first half of the century:

Soldiers of Salamis by Javier Cercas - set during the Civil War, I believe. I haven't read it yet, but it's been very popular.

Nada by Carmen Laforet - set in Barcelona in the 40s. This one is particularly popular in English translation, so it should be easy to find.

The Family of Pascual Duarte by Camilo José Cela - set in Spain mostly in the period right before the Civil War

The Back Room by Carmen Martín Gaite - deals with a woman's memories, which include the Civil War. I think I saw that a copy was posted on here a few days back, though I could be wrong.

I second the suggestion for The Time of the Doves. It's a gorgeous novel. I can think of a few others, but these are a few of the ones that I know have English translations floating around. I've read them all in Spanish, so I can't vouch for the quality of the translations.