Does anyone have an opinion on the use of 'their' versus 'his or her'? I know you should use 'their' with plural verbs, 'his or her' with singular, but what about those times when using 'his or her' is clunky, when using 'his' on its own is glaringly incorrect (because you are referring to a mixed crowd)? What do you think about substituting 'their' in fiction, where the writing is somewhere between formal and informal?

This is my sentence, referring to a gathering of men and women:
Each participant approached the water bearer to have their/his or her hands rinsed.