  A practical method to compute the Riemann zeta function is presented. The
method can compute $\zeta(1/2+it)$ at any $\lfloor T^{1/4} \rfloor$ points in
$[T,T+T^{1/4}]$ using an average time of $T^{1/4+o(1)}$ per point. This is the
same complexity as the Odlyzko-Sch\"onhage algorithm over that interval.
Although the method far from competes with the Odlyzko-Sch\"onhage algorithm
over intervals much longer than $T^{1/4}$, it still has the advantages of being
elementary, simple to implement, it does not use the fast Fourier transform or
require large amounts of storage space, and its error terms are easy to
control. The method has been implemented, and results of timing experiments
agree with its theoretical amortized complexity of $T^{1/4+o(1)}$.
