Won't hurt to try but most of the older heirloom peonies won't grow back without eyes, some of the newer hybrids will, depending on their parentage. I'd plant the bigger pieces with no eyes horizontally around an inch or so deep in a marked area so you won't forget what's there. Some can take 2-3 years to form eyes so it's not a quick process. If anyone in your area has tree peonies and wants to try grafting that might be a better use if the pieces with no eyes are fairly large, just an idea....

Thanks very much for the advice lizbest1, much appreciated. I just didn't want to toss them and find out later that I could have done something to them to make them viable. I hunted the internet but couldn't find any definitive answer so this was great. Cheers..

There are some that grow from blind roots very easily like most of the coral hybrids. You're not losing anything other than a square foot or so of space and a few minutes for trying, worth it!

I received a gift from a friend who was moving (her non-blooming peonies were in neglected piece of property). I planted everything anyways (tiny, anemic roots), not expecting anything for awhile if at all, but was pleasantly surprised this spring - most, even the tiny, tiny root chunks, are blooming. The non-bloomers have nice foliage, so probably next spring for these.