So how does one throw away one's likes and dislikes?

Actually, you can't. To "throw away" anything is an act of volition, and therefore involves the thinking mind. To throw something out, you have to first decide that you don't want it (which is picking and choosing in itself), then actually try to discard it, which only reinforces one's preferences. You may have a "preference" to have no "preferences," but as long as that's your preference, well, there you are.

You can't throw away your likes and dislikes. You can only allow them to leave.

Which is zazen. Sitting quietly, with no objective or goal, the thinking mind quiets down and "mental formations" such as likes and dislikes eventually drop away, as do other dualistic concepts like "self-and-other," and "mind-and-body." This dropping away happens at its own pace and in its own time - there's no way to force it, for as soon as you start to think about achieving the goal of losing these concepts, they immediately come right back.

The problem is the underlying delusion that there is a self that contains these things, and that they can be somehow made "not self." Once you start down that path, you're only reinforcing these habits, not getting rid of them.

As Dogen said,
To study the way is to study the self.
To study the self is to forget the self.
To forget the self is to be enlightened by all things of the universe.
To be enlightened by all things is to transcend the distinction of self and other and to go on in ceaseless enlightenment forever.

Which sounds pretty grim if that means that one then has no likes or dislikes or no affections or passions. "Enlightenment" does not mean becoming an emotionless automaton. As soon as one gets off the meditation mat, the thinking mind kicks back in, with all of it's picking and choosing and concepts of self and others. However, all this is accompanied by the realization that these preferences are just the artifacts of the thinking mind, that these preferences are based on illusions, and that they have no enduring reality.

Which sort of levels the playing field and makes both the fortunes and misfortunes much easier to take. My football teams may win or lose next week, and I may become excited or disappointed, but I'll also know that it's all just an artifact of my thinking mind.

But don't take my word for it. See for yourself. Come join me this weekend at the Atlanta Soto Zen Center for a weekend-long meditation retreat (sesshin) and allow body and mind to drop away. The retreat begins at 6:00 pm Friday December 2 and ends Sunday December 4 at 11:30 am, followed by a celebratory lunch. Check out the web site for more details. I'll be there all weekend.