Honestly, the best thing to do is to not let yourself get crossfaced in the first place, as a good crossface can be a bitch to escape once it's fully locked in. If you do find yourself in that position, though, just remain calm, guard against the mount and arm attacks like you normally would, and wait for an opportunity to slip out.
Unless it's a tournament and you're down on points (or it's training and your partner is a douche), he isn't just going to hold you there all day. Eventually he's going to try to improve his position or attack an arm, and once he does you'll have your chance to escape.
A nice little technique I've been using very effectively is the hand climb. Say you are side mounted and he is on YOUR RIGHT. That means he is cross facing you with his LEFT ARM. I take my right arm and work into against my cheek. The best way I would describe it is that its the gesture people make when they have a toothache. Like you're rubbing your cheek. Then once my hand is on my cheek, I will crawl it and work it along the side of my head as it guides the top man's cross facing arm up and over my head like I'm trying to make a "comb over" motion with my hand.
Now that the cross facing arm is over my head, I pop it up and start working to get out. I know it's a huge pain when someone heavy and good has a solid cross face on, you simply can't move but this has helped me a lot.
Gustavo Machado showed me a method to remove the cross face that I've had some measure of success with.
You are in bottom side mount with a good frame. Let's say he is on your right side so your left forearm is across his throat and your right elbow is controlling his hip. While keeping the elbow on his hip, get your right hand behind the elbow that is cross facing you, don't push just yet.
Now straighten your legs and put both feet together and lift them off the ground a bit. You want to make the bottom half of your body into a sort of pendulum. Start swinging your legs right to left to: gain momentum, make a bit of space, and loosen the hold.
When you feel the time is right (sometimes it's after 1 swing, sometimes it's as the guy reacts, it all depends...) swing to your left, look to your left and duck your head, use the hand on his elbow to push it over the top of your head. Turn back into him and touch elbow to knee to create a good escape posture and continue the shrimp escape from there.
I sometimes do the leg pendulum by itself when he holds on tightly just to start loosening grips and bait a reaction.