Rinse the corned beef under cool water. Allow it to soak in cold water to cover for 30 minutes and then drain it. Place the meat in a large kettle with fresh cold water to cover; add the peppercorns, bay leaves, and celery tops. Bring to a boil and turn down the heat to simmer uncovered over low heat for 3½ hours, skimming off any fat that rises to the top. (At this point you can stop, allow the beef to cool in the broth, and refrigerate to finish the next day.)
Add the carrots, potatoes and turnips to the warm broth and meat and simmer for 30 minutes. Add the wedges of cabbage and simmer another 15 to 20 minutes or until all the vegetables are tender.
Slice the corned beef thin, across the grain. Serve on a platter, flanked by rows of the boiled vegetables.