Bomber Quest

A young man named Shounosuke sees a strange dream: a voice talks about mankind and its sins, and a beautiful girl begs him to save her... The next day, when he goes to school, an earthquake occurs, and he is miraculously taken into another world, where he meets the girl from his dreams. She says that girls in different dimensions are in grave danger, and Shounosuke has to save them...

[From Mobygames]