Reserving this post for an eventual guide.
I'm busy this week, but maybe by the end of next week I'll have something more to share.

A couple of teasers:

The King:

Cost: 0/-/-/- (-25 on death)
Best Tier: N/A
Rating: N/A

Every army must have a King, so it makes sense that we start with him. As the linchpin of your army, you must defend your King throughout the entire game. While his death is no longer an instant loss as in vanilla chess, losing your King is still a major blow, since you lose not only the 25 morale up-front, but also an additional 3 morale each turn for the rest of the game, which can quickly add up to a defeat.
That being said, as pieces get traded away, the King can be used more aggressively, since his ability to move one step in any direction means it can control a decent amount of territory, particularly against single minions (against multiple minions he needs more help). You do have to be more careful with him than usual, though, because unlike in vanilla chess there are a number of pieces than can "solo-mate" the King even without backup, such as Dragon and HauntedArmor, and the time limit known as "morale decay" means that the board may not get sufficiently empty to justify his joining the fray before one side or the other loses outright.

Pawn:

Cost: 0/1/2/3
Best Tier: Base or +1
Rating: Mediocre (2)

Pawns have a big weakness: they can only move straight forward, yet they capture diagonally, which makes it easy for the opponent to drop a piece right in front of one and exploit a hole in your formation. In normal chess, this leads to the players creating intricate pawn chains and using them as the terrain around which the major pieces must play. In CEO, however, there are many other types of minions, many of which can attack straight forward and threaten the Pawn without being in danger itself. This makes the Pawn a relatively sub-par minion.
Pawns do have two distinct advantages, however: they're cheap and they're fast. If you can make trades with them you will almost always come out a couple points ahead. Also, with a Rook or similar piece behind them, they can be pushed far forward in a single move, particularly at higher tiers, potentially forking a pair of the opponent's more valuable pieces. The +3 Pawn in specific can be promoted in a single move if the column it sits on is empty. Still, they're fairly limited in what they can do and will often come out behind against a stronger minion line.
One other note: in vanilla Chess, Pawns can promote to Queens, and a single promotion will often decide the game. CEO Pawns can only promote to Bishops, which are distinctly less powerful than Queens. There are many other minions with promotions, some better than a Bishop, others worse. Getting any promotion at all, however, is still a major advantage - just not a game-ending one (at least in this case).

Knight:

Cost: 5/8/10/12
Best Tier: Base or +2
Rating: Good (3)

Among the "normal" chess pieces, the Knight is the one with the most complicated movement: unlike the others, it does not move in a straight line. In CEO, however, wonky movement is par for the course!
Knights are relatively short-ranged, but they make up for this shortcoming by being able to jump over other pieces in the way. This makes them very good at exploiting weak spots in the enemy formation. In CEO, however, minion lines tend to be fairly robust.