We develop a general solution technique for a dynamically accelerating crack in a linear viscoelastic material, based on a transform method developed by Slepyan (see [Models and Phenomena in Fracture Mechanics, Springer-Verlag, New York, 2002]) for solution of dynamic elastic fracture problems. We review the elastic fracture solution method and then treat the viscoelastic mode III fracture problem for crack tip speeds less than the short-time shear wave speed. The analysis includes an exact, closed-form expression for the stress intensity factor for an arbitrary time dependent crack face traction. As examples, we apply this solution method to the Achenbach--Chao and standard linear solid viscoelastic models.