Yes, you can. And no, it doesn't ruin the taste.
I wouldn't have believed it myself if I hadn't listened to Craig Ballantyne (and watched his video).
I didn't have as much trouble with my blender as Craig did, but his advice is spot-on. Adding leafy greens like spinach or kale to your smoothies or blended drinks is a great way to get high volumes of plant nutrients into your daily diet without having to sit down behind a punch bowl-sized salad three times each day.
This is sure to be only the first in many discussions about "green smoothies", if you are anything like me (or most people) it will probably take a few times for you to get used to the idea of combining spinach in a blender with the rest of your smoothie.
Here's another video of Atlanta Falcons' tight end Tony Gonzalez using spinach (and a bunch of other veggies) in his blender drink: