The Radio Ice Cherenkov Experiment(RICE) located at the South Pole, is designed to detect the coherent broad-band radio Cherenkov radiation emitted when a high energy (10¹⁵ to 10¹⁸ eV) neutrino interacts with a nucleon in the ice. Observations have identified that Gamma Ray Bursts(GRBs) are possible sites for high energy neutrino production. We consider here GRBs which occurred in the years 2001 to 2005 inclusive during the operational times of RICE. Using GRB photon spectral data, we calculate the neutrino spectra predicted for these GRBs and the subsequent event number expected in RICE. We re-analyze RICE data in small time windows surrounding the GRB burst start times using a refined method involving by eye analysis of this reduced data set and find no neutrino events in the data set. Using the effective volume of RICE appropriate for each GRB we calculate neutrino flux limits for the GRBs. Although the flux limits are several orders of magnitude weaker than the expected flux, the RICE GRB neutrino limits are the only limits in the PeV to EeV energy range.