We present results from our extensive survey of chromospheric and coronal activity in lower main-sequence members of the Hyades open cluster. We have obtained $H \alpha$ spectroscopy of ninety-one probable cluster members, twenty-three of which lie within three fields for which we have deep ROSAT X-ray observations. Twenty of the twenty-three stars are detected. Combining these observations with data from the literature, we find that the mean ratio log ($L_\alpha / L_{bol}$) is roughly constant for $M_{bol} > 8$, with a substantial dispersion. The X-ray-to-bolometric luminosity ratio increases with decreasing $M_{bol}$, although the relation may flatten for $M_{bol} > 9$. Comparing this behaviour with stars in the Pleiades cluster, there are significant differences: first, the log ($L_\alpha / L_{bol}$) relation in the Pleiades increases for $M_{bol} < 10$, but appears to turn over sharply at fainter magnitudes, with the lower luminosity stars having nearly an order of magnitude lower $H \alpha$ luminosity; second, we find no evidence for saturation in the log ($L_X / L_{bol}$) ratio. However, none of the lower luminosity Pleiades stars with low $H \alpha$ luminosity have X-ray observations.