Can I cut dead branches from my raspberries in the summer?

Yes you can. These dead branches are probably floricanes that produced berries and then died. This is a normal part of the raspberry life cycle. You can, and should, cut the dead branches off.

Related Videos:
Pruning Dead Floricanes from Blackberries
Building a Blackberry Trellis

Related Resources:
Raspberry Planting and Care for Home Gardeners
Growing raspberries in the home garden
Growing Raspberries in Colorado Gardens