Stuck at home, unable to work during the COVID-19 crisis, I quickly drew up this board and pieces. My goal was to do this using only a Glowforge and the materials I had on hand. There's some assembly, but no painting involved. Assembly is very straight forward and reasonably quick. The colors come from the mix of materials (two different types of wood and clear acrylic). If I make another, I'll do it in either all wood or all acrylic. With limited materials and an uncertainty as to when I'd be able to get more, I just went with the random pieces I had on the shelf.

Supplies

The board and chess pieces are cut out of two different colors of 1/8-inch thick material. The checkers are cut out of 1/4-inch thick material. You'll also need some glue, appropriate to the material you choose.

Step 1: Cut All of the Pieces

3 PDFs are provided with all of the pieces needed. Cut all of them, but read through the following before cutting to ensure you use the right material.

square.pdf is just that. It's a square. This is the base of the playing board. I used 1/8-inch board. This isn't seen, so you can use whatever you have around for this. Color/material/thickness don't really matter.
chess_pieces.pdf is everything needed to make one complete set of chess pieces. Each chess piece is made up of 3 flat pieces. There are 2 pieces that make up the outlines of chess pieces that will slide into each other to form the 3D piece, and a circular base that each one fits on. You'll need to cut this PDF twice, in different colors so you have pieces for both players. Since the round bases aren't all the same size, there are letters engraved on them to indicate which piece each base goes with. (Those engraved letters will be hidden when the pieces are assembled. They will not be visible during play.) This is all cut out of 1/8-inch board.
checkers_board.pdf contains several sections that need to be cut separately, even though they're in the same file. Things to be cut separately are in different colors so they can easily be turned on/off as needed. This file contains the following:
1. a frame that sits on top of the square base. The small checkered squares will be held in place by this frame. This is 1/8-inch material.
2. 2 sets of small squares to place inside the frame. They are in the file as two different colors to indicate that they are to be printed/cut from materials of 2 different colors. 1/8-inch material.
3. 2 sets of checkers. Each checker is made of two different pieces, allowing them to be stackable. When looking at the PDF, you'll see a circle with an engraved crown on it, and an outer ring. This is cut from 1/4-inch material.
4. There's a ring that is not actually part of the board but is used to set the checkers when they are glued. This ring is 1/8-inch thick.

Step 2: Assemble All of Your Chess Pieces

Each chess piece consists of 3 laser cut pieces. Matching pieces have complimentary slots. In the photos, you can see how one rook piece has a slot on top. The other has the slot on the bottom. Slide those together, using a small bit of glue to hold them in place. The round base of the rook has an "R" on it. The bottom of the two crossed rook pieces will fit into the cut cross in the base. Again, use a small bit of glue to hold it in place.

Assemble each piece in the set in the same manner.

Step 3: Assemble the Checkers

Each checker is made of two pieces: an outer ring, and a center disk. The are vertically offset from each other by half. The outer ring is the lower part of the checker. The disk is the upper part. Line the inside of the outer ring with glue.

There is one solo, extra large ring that will be used to set the checkers. This extra ring is half the thickness of the checker pieces. Place the upper disc of the checker face down inside this ring. Now place the outer ring of the checker onto the disc. The larger ring will hold the checker ring at the right height on the disc while the glue dries.

Alternate approach:

Unless you have very fast setting glue, this process will take forever with only the one ring. You'll need to leave each checker until the glue is dry before doing the next one. You can cut more rings, or take longer strips of 1/8-inch material and line up all checkers at once so the outer ring is raised up to the right height.