It's probably that awful music that wakes me up every mornin

这句话是强调句,"it's...that." 结构.probably是副词,为了强调它所以放在前面,其实它修饰的是"awful music that wakes me up every morning(糟糕的音乐没早把我吵醒)"这整段话.
还原后：Awful music(主语) wakes(谓语) me(宾语) up every morning(时间状语) probably(状语).
翻译：(很有)可能每天早上都是糟糕的音乐把我吵醒的.
故从这种意义上讲,be动词后可以接副词.