I've had to give up brocolli stalks.
Hiro LOVES to munch on them -- but they're MY favourite part too!!!
For training, I use these:
Sniffers 101 All Natural Training Bits for Dogs reviews | Buzzillions.com. They're about the size of a quarter. I break them into pieces and use the little bits for tricks, or whatever. If I'm doing a training session I'll use 3 and break them into 12 smaller pieces.