Thank you for using Ask an Expert for help with the insects on your raspberries. This appears to be one of the lecanium scales, which commonly attack fruit crops, but also are found on many ornamentals.

These critters hatch out in early to mid-summer to mate and disperse to new locations.  You can remove or squish any you find, or, as you did, cut out the heavily-infested branches or. When you do your annual pruning of the spent fruiting canes in fall, many of the next season’s brood will be removed, too.

As far as natural enemies, some of the tiny parasitic wasps will happily feed on the scale throughout the year.

As for control sprays, during the growing season, a pyrethrin spray is recommended. Check for one that has the OMRI (Organic Materials Review Institute) certification. Also, consider an oil spray during the dormant season. At that time, there are no leaves present and good coverage of the scale overwintering on the stems is quite effective. Many of the oils are registered for use in organic gardening. 

For more information on many facets of berry growing, check out the Oregon Extension publication Growing Raspberries in Your Home Garden.

Have a good gardening year,