How do I gain piety with Okawaru?

I've decided to play more melee characters lately, and have been trying Okawaru for my god so far. However, I'm not quite sure what is considered a powerful monster for gaining piety through sacrificing corpses (or for just killing 'em too, but I have a kill em all playstyle lately so that's kind of irrelevant). I'd like to be able to determine which corpses are better used for food, and which ones can get me to the point of getting those sweet sweet god gifts sooner.

Okawaru is a war-craving deity. Killing things will make him happy. He also appreciates corpses, which you can sacrafice to him by praying on top of the corpses. (p)

The killing is easy: you're already doing that.

With corpses, I typically sacrifice everything that I'm not going to eat or use; with most characters, corpses serve no other purpose than for the food, and it would otherwise just rot, especially melee characters (you're not likely to want them for necromancy). (Some classes, however, might want to keep corpses around.)

If I've just eaten, I'll just sacrifice, but once I think I'm getting closer to "Hungry", I'll chop a corpse and keep the chunk until I get hungry. It's a bit of a betting game with your hunger: do I chop the guaranteed food now, and hope I get hungry before it rots, or do I wait, and hope I find food when I'm hungry? Typically, you can be a bit agressive here: the dungeon will throw monsters at you readily, and it isn't the end of the world if you have to fall back on perma-food because you were too aggressive. Remember, you have "Hungry", "Very Hungry", etc. to get through.

If possible, I also prefer small, clean animals (like bats, rats, adders) for food (no Nausea), and big nasty things to impress Okawaru. If you're only Hungry, and you have a adder corpse and a huge but contaminated corpse, eat the adder. No point in getting 5 chunks of meat and have 4 rot. (The wiki article doesn't mention monster difficulty having an effect. The source code seems to indicate Okawaru appreciates the corpses of difficult monsters more.)

An Amulet of the Gourmand makes hunger management with Okawaru very easy (just fill up your hunger, and sacrifice to Okawaru's content.) (Of course, you lose the amulet slot to this.)

I've also been known to pick up a corpse in a battle, run up the stairs, drop it, sac it for the piety, and then determine how to deal with whatever caused me to run up the stairs. Note that picking up the corpse typically costs you at least two turns (one to move onto the tile, one to get the corpse): you need to make it up the stairs alive for it to be worth it. In a pinch, you can also drag monsters up stairs in a similar fashion.

Not casting Hero and Finesse at every battle also causes you to not expend piety on those things, but again, you need to make it out alive for it to be worth it. As someone noted once, “If you find yourself in a dangerous situation, it's already too late.”

You can see the wiki page for Okawaru for this, but you might want to avoid it if you don't want spoilers. (But then, you are asking here.)