With my system, each player keeps track of their own initiative. Since action “speeds” determine the die you roll and there are no modifiers to the die roll, a player can simply leave the number they rolled face-up to remind themselves of their initiative count on the next round. Then at the start of each new round I count up from 1 to 12. When a player’s number is called they can jump in with a “that’s me,” and if there’s a tie with another player, usually one of them will say “you can go first.” Once all creatures on a given count have finished their turns, you resume the count until everyone has acted, and then start from 1 again.
The DM does need to keep track of monster initiatives. But if you want to keep that simple, you can group monsters by hit die and roll each group’s hit die once, having all monsters in that group act on that initiative count. If you do this though, I recommend giving ties between monsters and PCs to the PCs instead of going in descending Dex order (or whatever).