A great music festival is the The Bronze, Blues & Brews music festival. Over Labor Day weekend is Juniper Jam, a great showcase of regional musicians in the folk/rock tradition. I highly recommend both. Two important cultural festivals include the Maxville Gathering that celebrates the logging history in Oregon and the annual Nez Perce art show. One of the biggest and most celebrated festivals is the Pendleton Roundup in September.
If there’s a particular type of festival or one in another part of Eastern Oregon you’d like to know more about, please let me know.