First, it's crucial that this job be completed in fall as the plants begin to go dormant, not in spring as they prepare to bloom. In September, choose a spot in full sun (at least 6 hours per day), and prepare the soil with well-rotted manure and/or compost, working it in and loosening the soil down to 18 inches.

Dig the plants carefully to minimize root injury, and remove the foliage. If the plants are large, they'll probably do better if they're divided. Gently wash the soil off the large roots, then use a sharp, clean knife to divide the plant's crown. Make sure each division has 3-5 pink buds or healthy stems. 

Plant divisions at least 3 feet apart (peonies need good air circulation), and set each division in a hole so the soil level is no more than 2 inches above the buds on the root. If the peonies are planted too deeply, they may not bloom.

 

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As it is, peonies resent disturbance and may not bloom for several years after transplanting -- especially if the plants are old. Water them well, checking for settling to make sure they are not too deep. If rain doesn't fall, water them every week or so until the ground freezes. Use a couple of inches of mulch in winter to prevent freezing and thawing from heaving the new plants out of the ground.

If you must move them in spring or summer, dig the plants carefully, disturbing the large root ball as little as possible (dig a deep, wide hole to ensure this). Then plant the peonies as soon as you can, taking care to plant them no deeper than they were growing before.