Thanks everyone for the suggestions! Very cool! I felt a little lost so I decided to summarize them.
Shoe hitting the cable - 7 votes. Nope, it was among the first things I checked. I do have a feed bag on the handlebar that attaches to the forks, sometimes the attachment gets loose and hits the tire/fork, but no, not this time.
Saddle/seatpost - 4 votes. It might be a Brooks creaking - it does that sometimes, but I can tell the difference by the sound and the pattern.
Cranks need greasing - 3 votes. Seems quite possible, but I lack the cap removal tool. I've already ordered one, so this one is on hold.
BB needs proper reinstall - regrease - 3 votes. Same as above, ordered a BB removal kit, currently on hold.
Indexing - 3 votes. Nope, the bike did come with indexing out of whack (for some reason, some bike shops can't figure out how to adjust indexing properly - I mean, you
watch Calvin do it and do the same until you've memorized it, how hard can it be?), but I fixed that first thing, and now it's top notch.
Chainring screws - 2 votes. I didn't even think in this direction, this being a new bike, but just in case, went ahead and checked them - they are fine.
Chain issues (quick link improperly installed, or lubing issue) - 2 votes. Nope, everything seems to be fine with the chain.
Rear derailleur pulleys - 1 vote. An interesting idea, I might do that, although by the sound of it, it sounds unlikely.
Something (a piece of grit, a metal shaving) between the crank axle and the BB - frankly, this is my primary suspect, but as I said, I can't verify this just yet.
Loved the tip about keeping my teeth out of the way of the pedals, have been doing my level best to follow it. Alas, I did take the chain off and tried listening to the cranks - it was music to my ears, but I'm not sure it was a true representation of the real life scenario (i.e., there was no load).
As to the nature of the sound - well, imagine a spoke nipple hitting a hard surface. That's the sound. Definitely more of a ticking than creaking.