What significance did the "heart" possess in the ancient cultures contemporary with the biblical era? I've had as hard a time with the concept of the heart as I've had with the concept of the soul.  Currently I am thinking the heart is a reference to what your goals are and what you direct your attention to. However, I am not sure.

What does the Bible mean when it refers to the word "heart"? Did they only use it in the sense of a four-chambered organ that circulates blood throughout the body, or did they use it in other ways? How do emotions relate to it? How do thoughts relate to it? What other insight does the scripture provide with respect to the heart?
In ancient thought, at least with respect to the Hebrews, the "heart" (לב) was considered to be the locus of one's thoughts. Today, most might say that thoughts originate from the brain and/ or mind.

Nevertheless, the idea of "to think to one's self" is expressed by the phrase לאמור בלב, which means, "to say in the heart" (cp. Gen. 17:17). This proves the connection they gave between "thought" and "heart."

To circumcise the foreskin of one's heart is to remove all the excess and unnecessary thoughts from one's thoughts and life and to devote one's entire attention and self to serving God. The foreskin signified pleasure and excess.

Were there any specific verses you had in mind?

Wilhelm Gesenius covered the topic well in his entry on the word לב in his lexicon, pp. 507-508.