How to draw a three dimensional lattice diagram to find out the answer to this math question?

I'm not sure what a "three dimensional lattice diagram" is, but a lattice diagram, or tree diagram, will work.

You may leave out the first step.

Put a dot on a piece of paper. This represents a starting point where nothing yet has happened.

"Throw" the die the first time. There are six possible outcomes from this throw, so draw six line segments going out from the original point. Put the numbers 1 through 6 at the ends of these segments.

This represents the possible outcomes of the first throw. If the first throw were a one and a second throw were made, there would again be six possible outcomes. So, beneath the 1, draw six line segments going out and put the numbers 1 through six at the ends of these, just as before. Repeat for 2 through 6.

Repeat this for the third throw. This will have to be a fairly wide table, in two dimensions, because the bottom row will have 316 numbers.

To find the probability of observing two odd numbers, go down each branch and see how many odds there are. For example, the first path is 1 -1 - 1, three odds, so this will be a failure. The second path is 1 - 1 - 2, two odds - a success.

This will need to be done for all 216 paths, and the final ratio of successes to total outcomes should reduce to 3/8.

By making all the first throws a "ring" around the initial point, the second throw a "ring" beneath the first throw, and the third throw a "ring" beneath the second, this could be three dimensional, but all the paths crossing would confuse me. "Hasse diagram" on Wikipedia has some examples that may help illustrate this poor verbal description.

Lattice Diagram