If you wish a modern trivet, not much country style as my previous design, you can of course take the BrittLiv ingenious way (see her ible here) or make a square trivet following these instructions.

Step 1: Material and Design

As usual I started from available material, and in this case I had a short wood board with square section. So I designed the trivet with a fast sketch, and happy of the result I begun to cut the wood pieces with a circular saw.

Step 2: The Rough Components

To make the connections between squares any little cylinder will be good. I opted for little dowels used for boards junctions, they're perfect for my purpose, except regarding the colour.
Arrange the nine pieces in a square shape, and pay attention that adjacent borders match one each other. When you've found the best arrangement, you can mark any piece with a number and with the position of the connections. Pencil traces will disappear when you'll sand the surfaces.
Also mark the center of any hole with a little punch, in this way you'll be able to drill the piece side with no efforts.

Step 3: The Connections

Because I didn't like the wood colour of the connections, I decided to envelope the dowels in black little hot-shrinking tubes. This will add a better contrast between light wood pieces and dark connection cylinders.
The cylinders are 30 millimeters long, and the central third will be covered by the shrinking tubes, so any piece of tube is 10 millimeters long.

Step 4: The Holes

Now you can drill the holes, pay attention to make any hole 10 mm deep, this will simplify centering the cylinders when you glue them.
Since you'll glue any piece, the holes could be a little wider than the cylinders, so you've not to push them too tight in place.

Step 5: The Finishing

When every piece is ready, it's time to sand them to make a better feeling to touch. Use two or more sand paper grain types, so to reduce the grain after some passes, and refine the surface at every step. Since the numbers will disappear, keep the arrangement of the pieces meanwhile you sand each of them.

Step 6: Preparation to Glue

The boards are now ready to glue.
Great thinking: it would be nice to make the trivet with 32 squares of pale wood and 32 more squares of dark wood, and arrange them in a chessboard way!

Step 7: Gluing

I used vinilic glue right for wood, the green type is more fluid, and the red one is more dense. Choose the one you prefer, probably liquid type id more handy to let the glue flow in the holes. Clean each connection very quickly after the gluing, so to not have glue residual when glue is dry.
At the end warm up the shrinking tubes, so to remove any wrinkle between squares.

Step 8: Done!

Your new modern trivet is now ready! If you wish to add some rubber feet I suggest you to glue one for each square, so to distribute the pot weight on the full trivet.
Have a nice cooking, see you in the next instructable ;-)