In most of Southeast Asia the drivers are courteous and it’s easy to find tranquil roads.
Laos and Cambodia have very little traffic and even on the main highway or national road you’ll find peaceful cycling. Do be careful of going too far off the beaten track here though. Smaller roads may not be paved, can turn to muck in the rainy season and in the worst case may not even exist! Just because it’s on a map doesn’t mean the road ever made it past the planning stage.
Secondary roads are your friend in Thailand. Any roads with a 4-digit number should be very quiet. Sometimes you’ll have to take a main road but the Thai highway network is set up for scooters and other slow moving vehicles so there’s almost always a generous shoulder. Thai drivers are excellent about leaving plenty of space as they pass.
We wouldn’t recommend cycling into Bangkok (it’s undoubtedly possible but far easier and less stressful to put your bike on a bus or train). On the other hand, getting around the city by bike is feasible and even fun. There is tons of traffic but nothing moves very quickly and everyone is friendly.
Malaysia and Singapore have the busiest roads in the region, particularly going down the west coast of Malaysia. Unless you enjoy taking your life in your hands, don’t cycle into or out of Kuala Lumpur. It’s a mad network of motorways and not recommended.