Folau, however, still believes Moore is the right man to take them forward.
"Steve's a great leader for us, everyone respects him in the team," Folau said on Friday.
"After the loss in Wellington he was quite upbeat and positive in the changeroom and making sure everyone's trying to stay positive.
"He's certainly well-respected amongst the group, there's no doubts about that."
Folau also stepped in to defend Cheika, who has copped the brunt of the attacks on the team as Australian rugby comes to terms with a 14th straight Bledisloe Cup series defeat.
Labelling some of the commentary surrounding Cheika as "unfair", he said most of it should be directed at the players.
"Cheik is the type of person who'll take the full blame but we know as players, a lot of that is coming back on us as well," he said.
"We've got to take a fair bit of responsibility there, to take the initiative and try and turn things around."
The Wallabies have lost their past six Tests - an unwanted streak Folau says he's never had to deal with in his career as an NRL or rugby player, although he did in his brief AFL stint for then-strugglers Greater Western Sydney.
However Australia will be favoured to beat South Africa in next Saturday's Rugby Championship clash at Suncorp Stadium.
"It's pretty vital for us considering the situation we're in at the moment," Folau said.
Several Wallabies, including James Slipper and fringe utility Luke Morahan, will play in the National Rugby Championship this weekend before the team gathers in Brisbane on Sunday.
Morahan is a strong chance to face the Springboks after the departure this week of Adam Ashley-Cooper, who has returned to France for club duties with Bordeaux.
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