For good weather and festivals, late June
Indonesia (Tenganan, Bali) Usaba Sambah (Fistfighting, traditional costumes) June/July, 1 day.
Philippines (Balayan) St. John’s Pig Parade (Roast pigs with costume are carried around the town for the baptism ceremony) 24th.
Papua New Guinea (Port Moresby) Port Moresby Show (Traditional Dances) Mid-June.
Ecuador Corpus Christi Festival with folk dancing, music, art monitored by both Christian and Pagan gods.
Peru, (Cusco) Inti Raymi. An Inca festival packed with colourful dances and parades, set in and around one of the most beautiful towns on earth. One week ending June 24th. Watch out for your wallet though.
Thailand, Loei, Phi Ta Khon Festival (Spirit Dance), late June.
Or go for festivals only, not the weather! i.e. this is not the best season
Malaysia, Kota Belud (Sabah, Borneo), Kota Belud Tamu Besar (tribal gathering), May/June, 1 day, Sarawak (Borneo), Gawai Dayak (tribal gathering), late-May/June. Malaysia Pictures.
Mexico (Papantla de Olarte, Veracruz State) Fiesta De Corpus Cristi (rituals, dance) May/early-June. Mexico Pictures
India Hemis Gompa (monastery), Hemis Festival (Tibetan monks festival), June/July, 2 days
Puri, Rath Yatra (Chariot Festival), full moon day in June/July. India Pictures.
Korea, Nationwide, Tano Festival (parade, wrestling, fireworks, floating lanterns+) May/June, 20 days
Chonju, P’ungnam Festival (dance, folk arts, games and sports), May/June, 1 day
Venezuela, Curiepe and the Barlovento region, The Big Drums of San Juan (Caribbean dancing), late June