The theory of phonon-drag thermopower S_g is given for a quasi-1D electron gas in a GaAs heterostructure. The coupling of the electrons to 3D phonons is considered through acoustic deformation potential and piezoelectric fields. An expression for S_g is given which may be used for quasi-1D wires of different geometries. Numerical results are presented for both the screened and the unscreened value of S_g for cylindrical wires in the temperature range 1-10 K. For temperatures T>or=2 K the major contribution to S_g comes from the deformation potential scattering but for T<2 K the contribution to S_g due to piezoelectric scattering is significant. Screening reduces S_g by 40%. S_g increases as the density of electrons decreases and curves of S_g/T³ against T show maxima. The overall behaviour is dominated by the 3D character of the phonon system and is similar to that found previously for a quasi-2D electron gas coupled to 3D phonons.