Similarly to what Kristin does, I don't cut greens, but I cook sweet potato, squash, etc. I usually leave them in ~3cm strips, and that's including things I don't cook usually like carrots or cucumber.
Funny story just came to my head: The other day I decided to give Ralph a whole grape, as I've read other people doing that. My mom was all concerned because it was big, and it was funny because she usually gives me a hard time for taking time to prepare Ralph's meals. I said "Haha! You LOVE Ralph! ^_^" And she's like "... No... ~_~;"
Anyhow, long story longer... I gave Ralph the grape, and left him to it. I came back in about a minute and it was already half gone. I watched him take a bite, then tear the rest away with his claws. What he didn't realize was that it had stuck on his claws, so when he finished his mouthful and went looking for the rest of the grape, he was swimming around with it on his claw! I bet he thought it was pretty nifty to have "food at hand" like that
