I have just such a sink/drainer.
First of all, think read hard and make sure you can't afford just a little granite around the sink!
If it has to be cheap, then wood is good. Best bet is Teak or Iroko, as these are much more moisture resistant. I have beech, as it was a colour thing, it's okay if oiled properly, and hard as nails, but you do need to take some time to oil it.
I find I get a lot of standing water in on the top, and you are forever wiping down.
Personally, I would hire a carpenter for the day to do it, as 40mm beech is hard work on you and your tools. Also, allow plenty of space for expansion, so if you tile above, leave a 3mm gap under the bottom tile (I tiled on top of some hardboard which I later removed) and cover with decent silicone.
The other problem is, you need to silicone before you oil it, but you really need to oil it before you fit it, including inside the tap holes, underneath round the sink edge. You will also need a foil vapour barrier over a dishwasher if it goes under the work top.
I does look nice, and you can sand it and oil it to look like new once and a while.
Paul