One thing to keep in mind--all these patients are VERY used to people spending lots of time just talking to them - it's a big part of their therapy. So don't feel awkward! They're expecting you to ask to talk to them.
I agree that in the beginning it can be easier to sit down and play a game with them. Maybe an easy card game, where you don't have to think too much, and can talk as you play.
Another thing I found works well is drawing. Sit down with someone (or even a group) and ask "Do you want to draw with me for a little while?" Ask them, "Draw the animal you would be if you could be an animal." You'll get some really interesting animals, and when you talk to them about why they chose them, many will really start opening up - at least, that's what I found when I tried it.
Surprisingly, another thing that really worked for me was this. During some "down time" when patients are just relaxing, go up to a small group, introduce yourself, and say, "Would anyone like to chat with me for a little while?" You will almost always get at least one person who really wants to talk - sometimes more, and then you have to ask them to take turns!
I know that normally you are assigned one patient to work with, but once you have talked with them for a little while, just make sure that your instructor is ok with you talking to other patients. It shows initiative on your part, and plus, it's easier for you to talk with someone who really wants to talk to you.