I put tea in my shakes all the time. I like using matcha best (really finely ground green tea powder) which is usually pretty expensive, but you can find it cheap if you look for it. Amazon sells Vita Life matcha cheap, which I bought once and it tasted like crap, but was masked well in fruit smoothies.
I buy mine now from silkroadteas.com where I buy all my tea (it's in the green tea section. It sells for $7 per 1/4 lb. but a little goes a LONG way and it really tastes much better than that Vita Life stuff and the matcha powders I've found in asian markets.
Since I like drinking tea a lot though, I oftentimes will save the tea leaves after they steep and throw them in with my shakes later, and I've never tasted them in the shakes when I use white, green, or oolong tea leaves. All the good healthy stuff in the tea does not come out in the first steeping, and the leaves are perfectly edible.
I don't like the bitter taste of black tea leaves in my shakes, but the exception to that would be chai. I usually will break open a bag of chai tea in a small mug, let it steep in just a small amount of hot water for a couple minutes as I'm throwing vanilla protein powder and ice and stuff in the blender, then toss in the concentrated chai, leaves and all, and blend it up. Tastes sorta like an iced chai latte.
Anyway, you can just throw a regular tea bag or teaspoon of loose leaf tea into your shakes, you don't have to steep it. But for some reason, there is a big difference in how I feel using matcha vs. throwing some loose leaf in the blender. It's essentially the same thing, but I guess the fine grind of matcha is absorbed better because it gives me a really great kick, which is especially useful before working out but it's also nice first thing in the morning.
It's like a nice little wake up jolt without the jittery effects of coffee or the caffeine crash making me tired later that coffee does. GOOOOOD stuff.
You can also experiment with throwing some loose leaf herbal "teas" in with your shakes later in the day if you don't want the caffeine- I sometimes throw in some rooibos, honeybush, or yerba mate, which have a lot of health benefits too.
Basically, do whatever you like. Tea is great!