Desiring God ministries has done a great job of providing resources to help Christians memorize the Bible. One of my favorites is the Fighter Verse App, which is available for iPhone and Android. Here are three great articles to whet your appetite for memorizing the word and some practical advice for doing it.
Twelve Gospel Passages to Soak In by David Mathis
Why Memorize Scripture by John Piper
Meditate on the Word of the Lord Day and Night by John Piper (sermon)