A Sino-Soviet war would be the best bet. If you got both sides to avoid nuclear weapons, you could end up with a scaled-up Iran-Iraq war, with the Soviets crushing any Chinese attempt on Siberia and the PLA/guerrilla forces wearing down Soviet advances into the more populated regions of Manchuria. Add to that an India/Pakistan conflict and maybe Vietnam as well and you could get an Asiawide war.
The only problem is naming. As the West would not be directly involved, occidental observers might very well be hesitant to call it a world war. The conflict might end up being known as the "East Asian war" or "Asian war" or simply "Sino-Soviet war" instead of WW3. I feel you'd have to involve more countries, especially nations closer to the West. Perhaps we could have something big going in the Middle East simultaneously, with one or more players there having friendships with the Indians/Soviets or Chinese/Pakistanis.