Technically, and formally, it should be "their being heavy1". However in normal speech, both them and their are equally acceptable.
1about is a preposition and needs a substantive as an object - "their being heavy" is a noun phrase.
The alternative is that "them" is the object of about. Although "They complain about them" is correct, this that leaves "being heavy" stranded and the sentence would have to end "... because they were heavy."