What are on each of the faces of the dice in the US version of Yu-Gi-Oh! Dungeon Dice Monsters?
What are on each of the faces of the different dice?


I found this Dungeon Dice Monsters page on the Wiki. There, it is specified each of the possible symbols and their meaning. The specific term used in the game is 'Crest', and you roll the dice to gather specific kinds of Crests to interact with the game.

Edit: I also found the official ruling in case it helps as reference. 

Worth mentioning that there are 4 kinds of dice, one for each possible monster level (1-4). Each dice has faces depending on the level they are (usually, the higher the level the less Summon Crests on the dice).

To summarize it here, the possible Crests are:


Summon Crest (star symbol): These crests are used to summon monsters. There are 4 kinds of monster levels, and thus Summon Crests for each level. The higher the level the harder is to achieve a summon. This is because Lvl 1 dice have 4 sides with Summon Crests, but Lvl 4 dice only has 1 side with Summon Crest.
Attack Crest (swords symbol): This Crest enables summoned monsters to attack. You need one Crest per attack, and a monster can only attack once per turn.
Defense Crest (shield symbol): This Crest helps monsters to defend themselves from attacks. Basically, the Defense Crest subtracts Attack from the attacking monster and then calculation proceeds.
Trap Crest (umbrella corp-ish symbol): Used to activate some monsters' effects when attacked.
Spell Crest (thunder symbol): Similar to Trap Crests, these are used to activate monsters' abilities.
Movement Crest (arrow symbol): Used to move your monsters across the board. One Movement Crests can move a monster one square (unless it is a Flying monster, which use 2 per square).


Regarding the content of the dice faces. On this wiki page we have, emphasis mine:


  No matter what version is played, the amount of summon crests is the same for all dice across a given level. Level 1 dice have four summon crests per die, Level 2 have three, Level 3 have two, and Level 4 have one. They all have other crests on the remaining sides.


Here is an image taken from the linked ruling booklet that specifies what is the content of each face of each dice level:



In text:


Lvl 1 Dice (Red): 4 Summon Crests, 1 Movement Crest, 1 (x2) Defense Crest
Lvl 2 Dice (Blue): 3 Summon Crests, 1 (x2) Movement Crest, 1 (x2) Attack Crest, 1 Spell Crest
Lvl 3 Dice (Yellow): 2 Summon Crests, 2 Movement Crests, 1 Attack Crest, 1 (x2) Trap Crest
Lvl 4 Dice (Green): 1 Summon Crest, 1 (x2) Movement Crest, 1 Attack Crest, 1 Defense crest, 1 (x2) Spell Crest, 1 Trap Crest.