We present a search for CO(1->0) emission in three Local Group dwarf irregular galaxies: IC5152, the Phoenix dwarf, and UGCA438, using the ATNF Mopra radio telescope. Our scans largely cover the optical extent of the galaxies and the stripped HI cloud West of the Phoenix dwarf. Apart from a tentative but non-significant emission peak at one position in the Phoenix dwarf, no significant emission was detected in the CO spectra of these galaxies. For a velocity width of 6 km/s, we derive 4sigma upper limits of 0.03 K km/s, 0.04 K km/s and 0.06 K km/s for IC5152, the Phoenix dwarf and UGCA438, respectively. This is an improvement of over a factor of 10 compared with previous observations of IC5152; the other two galaxies had not yet been observed at millimeter wavelengths. Assuming a Galactic CO-to-H_2 conversion factor, we derive upper limits on the molecular gas mass of 6.2 x 10^4 M_sun, 3.7 x 10^3 M_sun and 1.4 x 10^5 M_sun for IC5152, the Phoenix dwarf and UGCA438, respectively. We investigate two possible causes for the lack of CO emission in these galaxies. On the one hand, there may be a genuine lack of molecular gas in these systems, in spite of the presence of large amounts of neutral gas. However, in the case of IC5152 which is actively forming stars, molecular gas is at least expected to be present in the star forming regions. On the other hand, there may be a large increase in the CO-to-H_2 conversion factor in very low-metallicity dwarfs (-2 <= [Fe/H] <= -1), making CO a poor tracer of the molecular gas content in dwarf galaxies.
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