  We have used the 2- year Differential Microwave Radiometer data from the
COsmic Background Explorer (COBE) satellite to systematically search for
millimetric (31 - 90 GHz) emission from the Gamma Ray Bursts (GRBs) in the
Burst And Transient Source Experiment (BATSE) GRB 3B catalog. The large
beamsize of the COBE instrument (7 degs FWHM) allows for an efficient search of
the large GRB positional error boxes, although it also means that fluxes from
(point source) GRB objects will be somewhat diluted. A likelihood analysis has
been used to look for a change in the level of millimetric emission from the
locations of 81 GRB events during the first two years (1990 & 1991) of the COBE
mission. The likelihood analysis determined that we did not find any
significant millimetric signal before or after the occurance of the GRB. We
find 95% confidence level upper limits of 175, 192 and 645 Jy or, in terms of
fluxes, of 9.6, 16.3 and 54.8 10^{-13} erg/cm^2/s, respectively at 31, 53 and
90 GHz. We also look separately at different classes of GRBs, including a study
of the top ten (in peak flux) GRBs, the "short burst" and "long burst" subsets,
finding similar upper limits. While these limits may be somewhat higher than
one would like, we estimate that using this technique with future planned
missions could push these limits down to \sim 1 mJy.
