Turn off all loads on the circuits in question.
Turn off the main for this panel. Possibly overcautious, but that's part of what's let me get old.
Either: short the dead circuit, or turn on an incandescent bulb on it (main off, circuit dead, still dead - just the bulb/fixture/lamp turned on.)
Pull neutrals and look for one with low resistance from the hot in question. There should be only one. Unshort the circuit or turn off the lamp and make sure the resistance changes (if nothing else is turned on, should be infinite, but if there are things that are not able to be switched off, may be non-infinite. But you should see a difference when you switch the incandescent lamp off or remove the short - it's a form of verification.
I think you have enough neutrals not to worry too much about this, but: beware of "Multi-Wire-Branch Circuits" where one neutral serves two hots.
As for the wire colors, only white and green (with or without yellow stripe) are reserved (and IIRC, you can retag white to another color, but you can't retag green - and you can't retag anything else to white.) If your wiring is run in conduit, using the full range of available colors makes figuring out (what is connected to) the hot side of circuits much, much easier. Most household wiring is not run in conduit, so it's not often seen at home.
If the old fixture worked, at least one of the white wires is hot.
The easiest way to check which is hot is to use a non-contact tester. If the fixture is already removed, leave the two wires bundled together with a wire nut over the end. Put a wire nut on the single white wire. Turn the breaker back on. Only one of those should register on tester. That is the hot side. The other should not register. That is the neutral.
Once you know which wire is hot, mark it with black tape or a permanent marker.
If the the fixture is still attached, you can do the same thing with the fixture hanging down. Once you know what is what, turn the breaker back off before wiring.
The black wire goes to the hot wire, the white to the neutral. The ground wire should be attached to a green or bare wire if there is one in the box, or to the box itself if it is metal. If not, just cap it.
Best Answer
You need to pay attention to the comment by @Hari Ganti. The only way I know to make sure you are connecting the right wire to the right place with confidence. Would be to open up and inspect the connection point of the power supply. Then you could ring each wire with your ohmmeter, mark and install them correctly. But if you could get to your connection point, why wouldn't you install a manufacturer approved cord and avoid any complications?