I am not as experienced at this as FL-Flinter, above, but what he says conforms to everything I've been told by experts, and what I've followed as a practice in refinishing about 20 pair of grips over time, with great results.
I soak them in acetone for a day or so depending upon what it takes to dissolve the original finishing compound. Thick urethane ones take longer. Put them in a shallow tupperware type container with a cover, as it will evaporate quickly if you don't. Shiny urethane type coatings take the longest - scrubbing them with a toothbrush helps break up the coating and remove it faster.
I use Tung Oil, and the drying time is long but the results are excellent. It may not be practical on a large scale to use Tung oil, but since you are just doing one pair and care about the results, it's worth it. I rub it in by hand and allow to dry a day or two. I repeat this a few times to get the luster I want. Tung oil comes in high gloss and low gloss formulations from Formsby, so get the low gloss if that is what you are after.
Here's a pair of grips that were trashed up when I got them - peeling, cracked nasty urethane, wood all dull and boring. Now it really pops - and this is just Tung oil, not any urethane or anything. This is high gloss type.