We report on the first search for atmospheric and for diffuse astrophysical neutrino-induced showers (cascades) in the IceCube detector using 257 days of data collected in the year 2007-2008 with 22 strings active. A total of 14 events with energies above 16 TeV remained after event selections in the diffuse analysis, with an expected total background contribution of $8.3pm 3.6$. At 90% confidence we set an upper limit of $E^2Phi_{90%CL}<3.6times10^{-7} GeV cdot cm^{-2} cdot s^{-1}cdot sr^{-1} $ on the diffuse flux of neutrinos of all flavors in the energy range between 24 TeV and 6.6 PeV assuming that $Phi propto E^{-2}$ and that the flavor composition of the $ u_e : u_mu : u_tau$ flux is $1 : 1 : 1$ at the Earth. The atmospheric neutrino analysis was optimized for lower energies. A total of 12 events were observed with energies above 5 TeV. The observed number of events is consistent with the expected background, within the uncertainties.
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