The easiest way to germinate and grow peony seeds, is to plant them outdoors as soon as they are ripe, or in the spring if you obtain seed in the fall or winter. Some will emerge the spring after planting, although some will take an additional year.
Starting seeds indoors can speed germination; if you start early enough in the fall, you can have young plants growing in the spring. The method that I have successfully used is adapted from an article by Don Hollingsworth published in the Bulletin of the American Peony Society, and is outlined below:
1) Soak the seed in water for several days, changing water once or twice a day. This step hydrates the seed, and may help remove germination inhibitors. Discard any seeds that are mushy or become moldy.
2) Place the soaked seeds in a plastic bag with barely moist vermiculite or peat moss, and close tightly.
3) Incubate the seeds by placing the bag in a warm area, preferably 80 to 90 F. It is important that the seeds be exposed to this relatively high temperature. If held at room temperature, incubation will take much longer, and will not be as uniform. Thus it is important to find a spot with the right temperature, and verify it with a thermometer. Places I have successfully used include on top of a hot water heater, and above grow light fixtures.
Inspect the seeds every two weeks to check for the proper level of moisture in the bags, and check the condition of the seeds. You will probably not see any changes in the seed during the incubation period. The length of the incubation period will depend on the temperature, but normally three months is sufficient.
4) After the incubation period, move the bag to a cool location for rooting to occur, between 50 to 60 F. Again, this temperature range is important. If too cool, rooting will occur slowly. If too high, it will not occur at all. Roots will start to appear within a few weeks, and the roots can grow to six inches or more.
5) After roots have formed, move the seeds to a temperature of approximately 40 F. This step overcomes the leaf dormancy. Again, if the temperature is too cool, the process will occur more slowly. When dormancy has diminished, a leaf will begin to emerge. At this stage, the seed is ready to be placed in a warm area to grow. If removed from this step before dormancy has been overcome, they generally will not grow when planted out, so make sure a leaf can be seen emerging from the root. At this stage, the germinated seeds can be treated like any other growing plant, and planted outside or grown on in pots. The large root will require that a deep pot be used if growing on in ports. If removed from this step before dormancy has been overcome, they generally will not grow when planted out.
6) The first season, the plants may only produce a single leaf. Provide them with fertile, well drained soil, keep them well watered, and protect them from hot sun. In the fall, apply a covering of mulch in cold climates. The plants will get larger each year, with some blooming the the third year, some in the fourth year, and the stragglers in the fifth year.
For more information about starting seeds, please visit the American Peony Society website, or the website of the La Pivoinerie D'Aoust Nursery, which has great pictures of germinating peony seeds.