  Recent models which describe our world as a brane embedded in a higher
dimensional space introduce new geometrical degrees of freedom: the shape
and/or size of the extra dimensions, and the position of the brane. These modes
can be coherently excited by symmetry breaking in the early universe even on
``mesoscopic'' scales as large as 1 mm, leading to detectable gravitational
radiation. Two sources are described: relativistic turbulence caused by a
first-order transition of a radion potential, and Kibble excitation of
Nambu-Goldstone modes of brane displacement. Characteristic scales and spectral
properties are estimated and the prospects for observation by LISA are
discussed. Extra dimensions with scale between 10 \AA and 1 mm, which enter the
3+1-D era at cosmic temperatures between 1 and 1000 TeV, produce backgrounds
with energy peaked at observed frequencies in the LISA band, between $10^{-1}$
and $10^{-4}$ Hz. The background is detectable above instrument and
astrophysical foregrounds if initial metric perturbations are excited to a
fractional amplitude of $10^{-3}$ or more, a likely outcome for the
Nambu-Goldstone excitations.
