For a more colorful relish, use a combination of red and golden beets. Serve the pickled beets with poached or smoked salmon, or toss them into a salad made with toasted walnuts, endive, and fresh goat cheese.
Place beets in a large nonreactive saucepan, and add enough cold water to cover by about 2 inches. Bring to a boil over high heat; reduce heat to medium, and cook until beets are easily pierced with a paring knife, about 45 minutes.
Transfer mixture to a colander, and let drain, reserving 1 cup cooking liquid. Let beets cool slightly, then peel. The beets will be easier to peel when still warm. Cut each beet into quarters.
Combine vinegar, reserved cooking liquid, sugar, salt, and mustard seed in a large nonreactive saucepan. Bring to a boil over high heat, and continue boiling 2 minutes. Remove from heat, and add beets; let stand, covered, overnight at room temperature.
Transfer mixture to jars or airtight containers, and store, covered, in the refrigerator up to 2 weeks.