At this point in life the answer is usually that I do nothing other than call up memories of previous cultural consumption. If you are not at that point, Wikipedia is an excellent source for fiction and movies from a country. When it comes to music, consult the various Rough Guides to music; I mean the books, not the mediocre CD collections or the so-so travel guides. Also try the AllMusic guide, either paper or on-line; when it comes to music neither Amazon nor Wikipedia is to be trusted ("why not?" is an interesting question, is it because too many people feel entitled to have an opinion about music?). Bring music on cassette, CD, or iPod, as soundtrack for your trip, and ask your driver to put on Radio East Africa. Finding the best non-fiction books is the hardest category to master. I still prefer shelf browsing at libraries and book superstores.
An MR request is another option. Matt Dreyer asks what I recommend for a trip to Greece and Turkey. Offhand I’ll say Herodotus, the usual Greek classics, Pamuk’s Snow and Istanbul books, Sarkan (a Turkish singer), Sufi music, Greek traditional music from 1930-1950 (there are some wonderful collections, look for the word rembetika), a study of Turkish and also Greek textiles, a picture book on Cycladic art, a book on Greek sculpture at the National Museum in Athens, Norwich on the Byzantine empire, Michael Grant on the ancient world, Lord Kinross on the Ottoman centuries, a biography of Ataturk and there are a few good recent books which survey contemporary Turkey.
Your tips, either general or specific, are of course welcome.