I have found that presses make an excellent tea, though I have only had experience with French presses. In particular, French presses make a delightful mint tea, when you take an average press and use two teaspoons of green tea and three spear/peppermint sprigs. Delightful, I say.
Though the main reason I haven't had problems with presses is because I only press down the leaves far enough to keep them from coming out with the tea- don't press the leaves. Contain them. When you've poured the tea out, let it approach a better drinking temperature so that it's more mellow, and use that time to get the leaves out.
Simple as that.