Achromatic devices have wide application prospects in radar and imaging fields. However, chromatic aberration and limited bandwidth restrict their development. Moreover, broadband and highly efficient achromatic devices working in transmission mode are still difficult to realize. In this paper, broadband highly efficient achromatic transmission in the microwave region by a metasurface is achieved. First, the ideal dispersion conditions of achromatic meta-atoms are given. Then, a polarization selective grating metasurface and a split ring slot metasurface are designed using the transfer matrix method and equivalent circuit theory, respectively. The former is used to control phase characteristics while the latter enables controlling dispersion. Phase and dispersion can be controlled independently by cascading them and any phase curve can be designed as is desired. In order to verify the strategy, an achromatic deflector and an achromatic lens are designed and samples are fabricated. The experimental results show that the deflector can realize achromatic refraction from 9.3 to 12.3 GHz with average efficiency 77.5% and the lens can realize achromatic focusing from 9.8 to 12.2 GHz with average efficiency 78.9%, respectively. The experimental results are in good agreement with theory. The findings provide valuable strategy for achromatic devices design, which can be widely applied.