Is it OK to install hardwood directly on top of joists if the hardwood is perpendicular to the joist direction and the joists are spaced sufficiently close to one another? say 12" or 16" on center?

I understand that's how it was done many years ago. They did not use a plywood subfloor.

I wouldnt myself unless the floor is over 1" thick like those pine floor/ceiling combos you see in cabins. A normal 3/4 floor without a subfloor is going to bounce like a rubber ball if you lay directly over the floor joist. They didnt use a plywood in older homes it was a 1x4 on a diagnonal then a finished floor went over that. Some people mistake old subfloors for finished floors.

Just like sleepers on slab. I never figured out why that procedure was used. First one I saw or walked on was the same as Kevin mentions...more like a trampoline...and talk about noisy! Of course this one was about 15 years old so it lost all of it's integrity. Sleepers could have been coming loose as well. Too long ago to recall exactly. My first year in the business.

So, it sounds like it's not a great idea. If I need to minimize the increase in height, what other things might I do? For example, can I cheat on the plywood thickness?

Follow the mfg's instructions faithfully or you may regret it. They are in each box or you can download from most mfg website.
NWFA Inspectors are instructed to follow mfg's guidelines if they are more stringent than the NWFA'S. Which most mfg's advocate.

I've had to install over 1/2" plywood sub floors and I'm not a fan of it . 3/4" is the industry standard for normal sub floors.If you feel like you can't go thicker than 1/2" on your subfloor , it will still be better than going directly over your joists.

I have only seen this one time in a real old house and the floor moved and squeaked and have gaps.

There's virtually thousands of old floors done this way here in Tucson. Mostly with quartersawn oak and fir. But here the crawl spaces are dry as an Egyptian mummy. Some squeak, some are still solid as a rock. Most are 12 and 16 inches on center.
Try and tell these people who want to replace them that according to the NWFA you need plastic underneath and a plywood subfloor. You'll be tossed out of the house on your ear.