Well I've done it. I used double stick tape and fastened the wood to a scrap blank. Speed is not critical as long as you don't go to fast. To slow is simply not an issue, you just have to cut slower. The parting tool kerf can be an issue when you get to the smaller rings. It can bind up although on a 1/4" thick piece that shouldn't be a problem. I had problems because I was parting 3/4" pieces.
I've also done it where I just fastened the middle and when I parted through the wood comes off. In that case it was nice not running the lathe so fast and I had to wear gloves because the wood was kind of sharp when it came off. That's why I went to the double stick tape so the wood stayed in place until I was done. Just a couple of strips work fine.