The best time to transplant is fall.
The reason is, peonies LOVE to grow their HUGE root systems in cool weather. And a larger, healthy root system will grow a larger, healthier, better-blooming peony.
When the large roots with the eyes are transplanted, smaller roots grow out from these and the final root system will max out on a mature plant at approximately 3 feet x 3 feet.
In the fall, the ground is cooler, yet warm enough underground for several months, and the root system gets well-established by the time spring bloom season arrives.
If they are transplanted in spring, the months of summer heat really stresses out the plant. The large root system is much harder to establish. Some peonies, I've read, can take up to two years to bloom after spring transplanting.

PS - I divided & moved my Duchess de Nemours in fall, and both new plants bloomed the very next spring.

I have purchased peony roots in spring from Sam's Club (the ones I got were much bigger and had more eyes than the fancy nurseries!!) and potted them up in in 3-5 gallon pots with a good potting soil. I put them in full sun on the patio where I could make SURE they were watered regularly all summer.
Then I planted them in fall; the root systems had completely filled the pots & were spilling out the drainage holes. Many got a few blooms the next spring! Also, I worked up the ground where I planted(3x3 feet) & amended the soil as recommended for peonies so the roots could really take off growing in good, loose soil.
That's a lot of work, but since these peonies can live a LONG time, I figure it's worth it!
So I guess if you move peonies in spring, if you make SURE they don't get dried out at all in the hot months, maybe spring transplanting would be ok.
But if I don't have them on the patio, I forget to water!

What soil amendments would you reccomend to plant peonies?

The advice I read & used is to mix a couple handfuls of bonemeal to the dirt in the bottom of the planting hole, and/or some 6-month slow-release fertilizer (but not the 3-month Miracle-Gro brand because they said it dissolves too fast & could burn the roots)and mix in some aged compost & manure. I added a 40 lb. bag of topsoil because my soil is yucky orange-red clay subsoil the builders left on top after removing every bit of decent top soil, even weeds won't grow in the stuff!!! Some recommend coarse sand for drainage, because my beds are on a slope so I figure the drainage is already OK. But I'm certainly no expert; this is just from what I've read. Sue

My mother is wanting to get rid of the plants in one of her flower beds. She has six peonies approx. 10 years old. I've tried talking her into waiting until fall, after reading all these posts, but she wants to start a new flower bed now and could care less about the peonies. I really want them to live and plan on splitting them up with my two sisters.
What can I do to get the best possible result when transplanting in spring?

If you mother insists go ahead dig and divide. It would be better to do it in the fall but I have dug peonies all year. The thing you must do is when you divide them is to allow the open surfaces to dry. Wash them and lay out to dry. The cut surfaces will become chalky. Then either plant or as mention in other posts plant in large pots until fall. My personal preference is to plant as soon as possible.

thanks maifleur,
I appreciate the advice.

cathy