I've raced bicycles and motorcycles, and experienced these wobbles a lot, on motorcycles it is often called the "death wobble."

It is important that your wheels be true, that the frame is properly aligned, and that the hubs are precisely centered in the dropouts. The front wheel should be balanced.

How to you balance a front wheel? By using a cycle computer, the kind which uses a magnet clamped to one of the spokes. Put your bike upside down, your front wheel will usually turn by itself until the heaviest part is pointed downwards. Attach the cycle computer magnet opposite the heavy side. Then spin the wheel. If the wheel stops with the magnet on the low side, move the magnet along the spoke toward the hub, the closer to the hub it gets, the less weight is felt at the rim. Ideally, when you spin the front wheel, it should never stop in the same place twice.

You can also use tape weights on the rim or spoke nipples, but as I use a cycle computer, I like the method I described above, and it also allows for very fine balancing.