  In the next decade gravitational waves might be detected using a pulsar
timing array. In an effort to develop optimal detection strategies for
stochastic backgrounds of gravitational waves in generic metric theories of
gravity, we investigate the overlap reduction functions for these theories and
discuss their features. We show that the sensitivity to non-transverse
gravitational waves is greater than the sensitivity to transverse gravitational
waves and discuss the physical origin of this effect. We calculate the overlap
reduction functions for the current NANOGrav Pulsar Timing Array (PTA) and show
that the sensitivity to the vector and scalar-longitudinal modes can increase
dramatically for pulsar pairs with small angular separations. For example, the
J1853+1303-J1857+0943 pulsar pair, with an angular separation of about 3
degrees, is about 10^4 times more sensitive to the longitudinal component of
the stochastic background, if it is present, than the transverse components.
