This paper explores the potential for statistical epoch of reionization (EOR) measurements using wide-field radio observations. New developments in low-frequency radio instrumentation and signal processing allow very sensitive EOR measurements, and the analysis techniques enabled by these advances offer natural ways of separating the EOR signal from the residual foreground emission. This paper introduces the enabling technologies and proposes an analysis technique designed to make optimal use of the capabilities of next-generation low-frequency radio arrays. The observations we propose can directly observe the power spectrum of the EOR using relatively short observations and are significantly more sensitive than other techniques that have been discussed in the literature. For example, in the absence of foreground contamination the measurements we propose would produce five 3 σ power spectrum points in 100 hr of observation with only 4 MHz bandwidth with LOFAR for simple models of the high-redshift 21 cm emission. The challenge of residual foreground removal may be addressed by the symmetries in the three-dimensional (two spatial frequencies and radio frequency) radio interferometric data. These symmetries naturally separate the EOR signal from most classes of residual unsubtracted foreground contamination, including all foreground continuum sources and radio line emission from the Milky Way.