In a large pot, boil the beets in plenty of water until tender, 40 to 60 minutes. Meanwhile, in a separate pot, boil the potatoes in lightly salted water for about 20 minutes, just until soft. Drain the potatoes and set aside to cool.
When the beets are easily pierced with a sharp knife, drain them and plunge them in cold water. When they have cooled enough to handle, peel and chop them into bite-sized pieces.
In a serving bowl, mix together the beets, potatoes, red onions, yogurt, dill, vinegar and pickles. Add salt and pepper to taste and chill for at least 30 minutes.