Should I worry about the fact that my newborn tries to suck on anything that brushes his cheek?

Absolutely not. This pattern of behavior is called the rooting reflex and is common in all healthy newborns. The rooting reflex part of a collection of primitive reflexes that all newborns exhibit. Primitive reflexes are necessary for the baby's survival in the first several months of life.

You can provoke the rooting reflex by stroking a baby's cheek with your finger or his mother's nipple; in response, the baby's head will turn toward the stimulus. The baby will then open his or her mouth and search, or root around, for nourishment. This reflex should disappear by the time the infant is 3 to 4 months old.