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)Φ_(90%CL) < 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 Φ ∝ E^-2 and the flavor composition of the ν_e : v_μ v_ τ 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.