Actually is the best time to travel in South-East Asia from October to March, but this is not completely right, as you can travel in south-east Asia the whole year. But the European winter month are perfect for this region. Except: South-east cost of Thailand and the east coast of Malaysia, there should you go in the European summer.

countries in details:

Thailand
You can travel Bangkok the whole year (around 30°C and high humidity).
Best time to travel for the west coast is middle of October to April.
The north of Thailand and the gulf of Thailand are good from May to September.

Malaysia Peninsular, east coast and central
Good: June to August «dry season» less rain.
Not so good: October to February. «Monsoon», hot and high humidity.

Malaysia Peninsular, west coast
Good the whole year, except north of Penang.
Not so good: May to Sept. around Penang. April to Oct. south west coast «Monsoon»

Indonesia
Bali is fantastic the whloe year around
Good: May to October «dry season» less rain.
Not so good: November to March. «Monsoon»
Java has a lot rain in January and February.
Also a lot rain is in November and December at the west coast of Sumatra.
North Sulawesi is great from May to October.

Singapore
Is good the whloe year around; the most rain is from November to January

Borneo
Good: March to September «dry season» less rain.
Kinabalu - National park: February to April best time for tracking
Not so good: November to February. «Maximum of rain»

Cambodia
Good: November to April «dry season» mostly no rain.
Not so good: May to October. «Monsoon»

Laos
Good: November to April «dry season» mostly no rain.
Not so good: May to October. «Monsoon»

Vietnam
Good: November to April «dry season» in the south and north.
The central coast and the mountains are good to travel the whole year.
Not so good: May to October. «Monsoon»  in the south and north

Myanmar
Good: middle of October to April. Hot from March.
Not so good: April to September. «Monsoon» really hot in May (up to 45°C in Mandalay).

Philippines
Good: December to May «dry season»
Not so good: June to November. «Monsoon» with storms.