The mixed group of dogs rushed about, vocalizing in a most distressed manner. Which's bark, or howl or moan, was the most disturbing?
Or, if you think that even generic dogs are human enough to require who, then:
The scientists had set up a display with half a dozen species of cockroaches. You could even put on a glove, stick your hand in the box, and let them crawl all over you. Which's touch was the most disgusting?
Note that while you could rephrase the question to avoid the word, that's not necessarily appropriate. For example, which was the most disgusting brings in questions of appearance, behavior, and even sound; if I want to focus on a particular characteristic like touch, the form in the example is better.
(I have bronchitis; in lieu of sleeping soundly, I get to think about things like this.)