Here's how I keep track of every card played in a hand of bridge, or any other card game, for that matter:
1) Think about it.
2) Use shorthand.
Let's first talk about number 2: using shorthand.
It's difficult to juggle more than seven numbers or items on a list in your head; so it's a bad idea to try to remember every card one by one as it's played. I don't keep a mental checklist with 52 boxes to check off. I just remember a single four digit number.
On each hand, I start off with 0000. That's how many cards have been played from each suit, in suit order (spades, hearts, diamonds, clubs). After each trick, the number goes up: 0400, 0701, 0741, and so on. Counting the cards in my hand, I pretty much always know how much is left of every suit. If a suit's count goes over 9, I'm fairly confident in remembering that there is only either one or two cards left in that suit; if there are three cards, I don't have any of them, so it generally doesn't matter to me, anyway.
That covers number. What about the values, such as the honor cards? Oddly enough, just doing the number count is pretty much all I need to do in order to remember the values played, too. When I think of the number (e.g. "7 hearts have been played"), I usually remember the tricks in which they were played, and therefore the values played in those tricks, too.
Which brings us back to number 1: thinking about it.
If I don't think about something, if I give something not a moment's thought, the memory of that thing passes through me like water through sky. I can walk by people I know, but if I don't pay attention, no impression of their passing remains with me. You can tell me your name, but if I don't take a moment - just a fleeting moment - to think about it, or remembering it, I won't even have heard you say it.
A moment's thought it no guarantee that I'll remember anything for a long time; interference, such as the unimportance of the thing or a cacophony of other things can crowd out the memory and cause me to forget, even when I take a moment to remember. To remember things long term, I have to repeat them to myself, write them down, or perform some other kind of conscious task. I couldn't tell you most of the bridge hands right after I've played a set of them, unless I write the results down after each hand.
A moment's thought is the difference between a short term memory and no memory at all.