We are carrying out sensitive X-ray observations with Chandra and XMM of type II quasars selected from the Sloan Digital Sky Survey based on their optical emission line properties . In this paper we present the first results of our program . We present observations of four objects at redshifts 0.4 < z < 0.8 and an analysis of the archival data for four additional objects in the same redshift range . Six of the eight were detected in X-rays ; five of them have sufficient signal to derive spectral information . All of the detected sources have intrinsic luminosities L _ { 2 - 10 keV } > 5 \times 10 ^ { 43 } erg s ^ { -1 } ; the five with sufficient counts for spectral fitting show evidence for significant absorption ( N _ { H } \gtrsim a few \times 10 ^ { 22 } cm ^ { -2 } ) . At least three of the objects likely have N _ { H } > 10 ^ { 23 } cm ^ { -2 } ; some may be Compton-thick ( N _ { H } > 10 ^ { 24 } cm ^ { -2 } ) . In the five objects for which we could fit spectra , the slopes tend to be significantly flatter than is typically observed in AGN ; it is possible that this is due either to reprocessing of the nuclear emission or to a line of sight that passes through patchy absorption .