  The opening of the gravitational wave window by ground-based laser
interferometers has made possible many new tests of gravity, including the
first constraints on polarization. It is hoped that within the next decade
pulsar timing will extend the window by making the first detections in the
nano-Hertz frequency regime. Pulsar timing offers several advantages over
ground-based interferometers for constraining the polarization of gravitational
waves due to the many projections of the polarization pattern provided by the
different lines of sight to the pulsars, and the enhanced response to
longitudinal polarizations. Here we show that existing results from pulsar
timing arrays can be used to place stringent limits on the energy density of
longitudinal stochastic gravitational waves. Paradoxically however, we find
that longitudinal modes will be very difficult to detect due to the large
variance in the pulsar-pulsar correlation patterns for these modes. Existing
upper limits on the power spectrum of pulsar timing residuals imply that the
amplitude of vector longitudinal and scalar longitudinal modes at frequencies
of 1/year are constrained: ${\cal A}_{\rm VL} < 4.1\times 10^{-16}$ and ${\cal
A}_{\rm SL} < 3.7\times 10^{-17}$, while the bounds on the energy density for a
scale invariant cosmological background are: $\Omega_{\rm VL}h^2 < 3.5 \times
10^{-11}$ and $\Omega_{\rm SL}h^2 < 3.2 \times 10^{-13}$.
