We present the methodology for ``blind millimetre-wave surveys for redshifted molecular absorption in the CO/HCO$^+$ rotational lines. The frequency range $30-50$ GHz appears optimal for such surveys, providing sensitivity to absorbers at $z gtrsim 0.85$. It is critical that the survey is ``blind, i.e. based on a radio-selected sample, including sources without known redshifts. We also report results from the first large survey of this kind, using the Q-band receiver on the Green Bank Telescope (GBT) to search for molecular absorption towards 36 sources, 3 without known redshifts, over the frequency range $39.6 - 49.5$ GHz. The GBT survey has a total redshift path of $Delta z approx 24$, mostly at $0.81 < z < 1.91$, and a sensitivity sufficient to detect equivalent ${rm H_2}$ column densities $gtrsim 3 times 10^{21}$ cm$^{-2}$ in absorption at $5sigma$ significance (using the CO-to-${rm H_2}$ and HCO$^+$-to-${rm H_2}$ conversion factors of the Milky Way). The survey yielded no confirmed detections of molecular absorption, yielding the $2sigma$ upper limit $n(z=1.2) < 0.15$ on the redshift number density of molecular gas at column densities $N({rm H_2}) gtrsim 3 times 10^{21}$ cm$^{-2}$.
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