A complete, scalable prototype analysis system for the observation of gravitational wave bursts with a network of interferometric detectors is designed and tested. This system detects localized transients in the gravitational wave data streams from two independent detectors using a time-frequency algorithm, applies veto conditions on these transients based on auxiliary channels, and looks for significant coincidences between the surviving transients from the two detectors. The analysis system was optimized for setting rate upper limits on three populations of astronomical sources, using preliminary data from the LIGO Project (the E7 Engineering Run data). The three classes of sources were core collapses, bar-mode instabilities in neutron stars, and equal-mass black hole binary coalescences, and a 95% confidence rate upper limit of 2 per hour was set, for sources uniformly distributed within a distance of 6 pc, 950 pc, and 3.2 kpc from the Earth, respectively. A detailed discussion of the character of the noise in the data used to derive these limits shows that a network of interferometers including the instruments of the GEO, LIGO and VIRGO Projects should attain a 50% detection efficiency for these sources out to distances of 40 kpc, 1 Mpc and 40 Mpc, respectively, for a false alarm rate giving expected upper limits of 6, 0.6, and 2·10⁻¹ per year per galaxy, respectively, for a full year of observation.