We present 30-GHz Sunyaev-Zel'dovich (SZ) observations of a statistically complete sample of galaxy clusters with the One Centimetre Receiver Array-prototype (OCRA-p). The clusters are the 18 most X-ray luminous clusters at z > 0.2 in the ROSAT Brightest Cluster Sample. We correct for contaminant radio sources via supplementary observations with the Green Bank Telescope, also at 30 GHz, and remove a cluster that is contaminated by an unresolved X-ray source. All 17 remaining clusters have central SZ effects with Comptonization parameter y 0 exceeding 1.9 × 10−4, and 13 are detected at significance ≥3σ. We use our data to examine scalings between y 0 and X-ray temperature, X-ray luminosity and the X-ray mass proxy Y X, and find good agreement with predictions from self-similar models of cluster formation, with an intrinsic scatter in y 0 of about 25 per cent. We also comment on the success of the observations in the face of the contaminant source population, and the implications for upcoming cm-wave surveys.