Axonemal dynein is the molecular motor responsible for the rhythmic beating of eukaryotic cilia and flagella. An individual axonemal dynein molecule is capable of both unidirectional and oscillatory motion along a microtubule (Nature 393 (1998) 711). We propose a model which links the physical motion of a two-headed dynein molecule to its ATP hydrolysis cycle, and which exhibits both processive and oscillatory behaviors. A mathematical analysis of the model is used to make experimentally testable predictions.