1. A magnetic resonance imaging apparatus comprising: a data acquisition unit configured to acquire from a patient plural pieces of data for imaging and plural pieces of projection data used to obtain respectively corresponding data related to respiratory motion of the patient by continuously applying a radio frequency pulse train to the patient, said pulse train having a constant repetition time (TR) interval, and both the plural data for imaging and the plural pieces of projection data being acquired asynchronously with cardiac time phase; a correction unit configured to perform motion correction of the plural pieces of data for imaging using the respiratory motion-related data obtained based on the plural pieces of projection data; a data sorting unit configured to sort the plural pieces of data for imaging after motion correction into a cardiac time phase order based on electrocardiographic information also obtained from the patient; and an image reconstruction unit configured to reconstruct three-dimensional cine image data based on the plural pieces of data for imaging sorted in cardiac time phase order after motion correction.