  Kinesin is a molecular motor that transports cargo along microtubules. The
results of many {\it in vitro} experiments on kinesin-1 are described by
kinetic models \cite{Clancy11} in which one transition corresponds to the
forward motion and subsequent binding of the tethered motor head. We argue that
in a viscoelastic medium like the cytosol of a cell this step is not Markov and
has to be described by a non-exponential waiting time distribution. We
introduce a semi-Markov kinetic model for kinesin that takes this effect into
account. We calculate, for arbitrary waiting time distributions, the moment
generating function of the number of steps made, and determine from this the
average velocity and the diffusion constant of the motor. We illustrate our
results for the case of a waiting time distribution that is Weibull. We find
that for realistic parameter values, viscoelasticity decreases the velocity and
the diffusion constant, but increases the randomness (or Fano-factor).
