An analysis of the decays of $B^\mp \rightarrow D K^\mp$ and $B^\mp \rightarrow D \pi^\mp $ is presented in which the $D$ meson is reconstructed in the three-body final states $K^\mp \pi^\pm \pi^0$, $\pi^+ \pi^- \pi^0$ and $K^+ K^- \pi^0$. Using data from LHCb corresponding to an integrated luminosity of 3.0 fb$^{-1}$ of $pp$ collisions, measurements of several $CP$ observables are performed. First observations are obtained of the suppressed ADS decay $B^\mp \rightarrow [\pi^\mp K^\pm \pi^0]_D \pi^\mp$ and the quasi-GLW decay $B^\mp \rightarrow [K^+ K^- \pi^0]_D \pi^\mp$. The results are interpreted in the context of the unitarity triangle angle $\gamma$ and related parameters.