Chia doesn't have much interest in Mulberry wood, but I think it's only because she's come to expect apple, and is disappointed (the same is true for shreddies versus any other treat.) If I leave the mulberry in the cage for a while, she'll eventually eat it.
She also loves anything she can carry. I find the pieces she tends to chew less are the ones that are heavy, or attached to the cage.
One treat she hates is hibiscus flower. I gave it to her once, and it went down like this: She grabbed it from me, sniffed it and made a weird face... and then dropped it and RAN AWAY. I picked it up and tried to give it to her again, and she kept running away. I left it in the cage for an hour or so, and when I came back it was shredded and she'd tossed half of onto the floor. I haven't tried again lol.