What part of speech is under, as in under the feather bedding.

Under is a preposition. 

Prepositions are words that indicate location. It tells the location of the subject or object. 

Prepositions not only show the actual location of a place but also the location in time.

Some prepositions of an actual location are: in, on, under, above, behind, beside, etc.

Some prepositions of a location in time are: before midnight, after the exams, during winter

Preposition is the part of speech