Since 2004 we have been carrying out a pulsar survey of the Cygnus region with the Westerbork Synthesis Radio Telescope (WSRT) at a frequency of 328 MHz. The survey pioneered a novel interferometric observing mode, termed 8gr8 (eight-grate), whereby multiple simultaneous digital beams provide high sensitivity over a large field of view. Since the Cygnus region is known to contain OB associations, it is likely that pulsars are formed here. Simulations have shown that this survey could detect 70 pulsars, which would increase our understanding of the radio pulsar population in this region. We also aim to expand the known population of intermittent and rotating radio transient (RRAT)-like pulsars. In this paper we describe our methods of observation, processing and data analysis, and we present the first results. Our observing method exploits the way a regularly spaced, linear array of telescopes yields a corresponding regularly spaced series of so-called ``grating'' beams on the sky. By simultaneously forming a modest number (eight) of offset digital beams, we can utilize the entire field of view of each WSRT dish, but retain the coherently summed sensitivity of the entire array. For the processing we performed a large number of trial combinations of period and dispersion measure (DM) using a computer cluster. In the first processing cycle of the WSRT 8gr8 Cygnus Survey, we have discovered three radio pulsars, with spin periods of 1.657, 1.099 and 0.445 seconds. These pulsars have been observed on a regular basis since their discovery, both in a special follow-up programme as well as in the regular timing programme. The timing solutions are presented in this paper. We also discuss this survey method in the context of the SKA and its pathfinders.