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 8.3 +/- 3.6. At 90% confidence we set an upper limit of E(2)Phi(90%CL) < 3.6 x 10(-7) GeV.cm(-2).s(-1).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 proportional to E(-2) and the flavor composition of the nu(e):nu(mu):nu(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.