We have investigated the dimensionality and origin of the magnetotransport properties of LaAlO3 films epitaxially grown on TiO2-terminated SrTiO3(001) substrates. High-mobility conduction is observed at low deposition oxygen pressures (P(O2)<10(-5) mbar) and has a three-dimensional character. However, at higher P(O2) the conduction is dramatically suppressed and nonmetallic behavior appears. Experimental data strongly support an interpretation of these properties based on the creation of oxygen vacancies in the SrTiO3 substrates during the growth of the LaAlO3 layer. When grown on SrTiO3 substrates at low P(O2), other oxides generate the same high mobility as LaAlO3 films. This opens interesting prospects for all-oxide electronics.