I will first give a brief introduction to T. Mochizuki's Theory of twistor D-modules. Then, we use it to study Kodaira-type vanishings. In particular, we will generalize Saito vanishing, and give a Kawamata-Viehweg type statement. As an application, we will also prove a Bott-type Vanishing using M. Saito's mixed Hodge module.