Question about prayer

Eliot Ravenwood’s recent (excellent) post about Scripture and Greek religion referenced in passing the Bible’s intstruction regarding “petitionary prayer.”

I have been researching lately the spirituality of the “Desert Fathers,” and their descendents, Anthony the great, Chrysostom, Benedict, etc., and their emphasis not only on petitionary prayer, which is a form of discursive or verbal prayer, but on non-verbal and non-discursive prayer.

My question is: “What does the Bible teach, if anything, about non-discursive prayer?”

I have some leads from the Psalms, perhaps, and inferences from the life of Jesus, but are there any explicit teachings on prayer that are not teachings on some form of discursive prayer, petitionary or otherwise?