  We present the results and methodology of a search for neutrinos produced in
the decay of charged pions created in interactions between protons and
gamma-rays during the prompt emission of 807 gamma-ray bursts (GRBs) over the
entire sky. This three-year search is the first in IceCube for shower-like
Cherenkov light patterns from electron, muon, and tau neutrinos correlated with
GRBs. We detect five low-significance events correlated with five GRBs. These
events are consistent with the background expectation from atmospheric muons
and neutrinos. The results of this search in combination with those of
IceCube's four years of searches for track-like Cherenkov light patterns from
muon neutrinos correlated with Northern-Hemisphere GRBs produce limits that
tightly constrain current models of neutrino and ultra high energy cosmic ray
production in GRB fireballs.
