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I've been eyeing some dunhill pipes with a really nice looking birdseye pattern and I'm wondering how I might achieve that myself. Is it a certain cut of the briar or is it more an orientation?

That's referred to as a cross cut from ebauchon.
These blocks show the birds eye on the sides.

In general, birdeye is "endgrain", whereas flame or straight grains are cut parallel to the grain of the wood. So if you want a pipe that exhibits birdeye on the sides, most ebaucons will accomodate that (some being far better than others). If you like the birdeye on the top, a plateux cut will treat you better.

Of course, you can cut a pipe in whatever direction you want in any given block (with certain size limitations).

Get yourself a really fine paintbrush and go to it.

...or buy a cross-cut, "ebauchon" cut block, or a plateaux block on its side.

That's sort of what I figured. Thanks