  Stellar mass compact objects in short period orbits about a
$10^{4.5}$--$10^{7.5}$ solar mass massive black hole (MBH) are thought to be a
significant continuous-wave source of gravitational radiation for the ESA/NASA
Laser Interferometer Space Antenna (LISA) gravitational wave detector. However,
these extreme mass-ratio inspiral sources began in long-period, nearly
parabolic orbits that have multiple close encounters with the MBH. The
gravitational radiation emitted during these close encounters may be detectable
by LISA as a gravitational wave burst if the characteristic passage timescale
is less than $10^5$ seconds. Scaling a static, spherical model to the size and
mass of the Milky Way bulge we estimate an event rate of ~ 15 per year for such
burst signals, detectable by LISA with signal-to-noise greater than five,
originating in our galaxy. When extended to include Virgo cluster galaxies our
estimate increases to a gravitational wave burst rate of ~ 18. We conclude that
these extreme mass-ratio burst sources may be a steady and significant source
of gravitational radiation in the LISA data streams.
