The electron spin dynamics is studied by time-resolved optical orientation experiments in strained InGaAs/GaAs quantum wells (QWs) grown on (111) or (001) substrates. For a given well width, the electron spin relaxation time in (111) InGaAs QWs decreases by an order of magnitude when the indium fraction in the well varies only from 4% to 12%. In contrast, the electron spin relaxation time depends weakly on the indium fraction in similar InGaAs quantum wells grown on (001) substrates. The strong variation of the electron spin relaxation time in (111) strained quantum well can be well interpreted by the Dyakonov-Perel spin relaxation mechanism where the conduction band spin-orbit splitting is dominated by the structural inversion asymmetry (Rashba term) induced by the piezoelectric field. In (001) QWs, due to the absence of piezoelectric field, the electron spin relaxation time is solely controlled by the Dresselhaus term. These results demonstrate the possibility to engineer the electron spin relaxation time in (111)-oriented quantum wells by the piezoelectric field induced by the built-in strain.