Some people claim card counting is one of the best strategies when playing hearts. While I can't either agree or disagree since I don't play hearts much at all, I've found out some basic information about how to count cards in hearts games.
Keep Track of Suits
A fairly basic thing to keep track of is the amount of times a suit has gone around. Even my understanding of the game is enough to know that everyone should at least try to keep track of this one.
How Many Cards of Each Suit Remain?
The next important detail to track is how many cards of each suit are left. If you can't keep track of all the suits at the same time, which is very understandable, you should at least keep track of the most important ones.
It's hard to be so good at counting cards at hearts that you could remember every single card played. That's why you should start with simple methods and move on to advanced stuff once you manage to keep track of simple counts.
I recommend reading Mark's Hearts Tips for more information on how to count cards at hearts.