I am looking forward to the good answers you will get, that board is really fine. But be aware, you won't get this classy look back, you will have to settle with that classy look, then. For my part, if the board is not outright sticky, the grime is the finish time gave it, so I keep it.
If it came with pieces, would you mind showing them on the board? If it did not have pieces, you could wait until you get matching pieces before deciding on a right cleaning procedure.
Thank you for the reply @ungewichtet. It did come with pieces:
It's an old Vienna coffeehouse set which, based on the knight, I think must date from the early 20th century. It is a very small set with the king only measuring 58mm.
The board and pieces match nicely so it may be a real "set" but that's just a guess (or wishful thinking maybe) Board is 31x31cm and is the perfect size for the small cupboard it will be placed on.
Perhaps something like …
Citrus Terpene (CT) and a cloth … a bit of CT on a soft clean cloth - it might remove a layer of dirt/grime but not the beautiful colour/patina and/or then CT and 0000 steel wool … if you wanted it slightly cleaner.
Also, I’ve now fallen in love with a bees wax and oil mixture applied with either a soft cloth or if I want to smooth some lite surface scratches - oooo steel wool and beeswax/oil.
Note here though that the wax oil mix might slightly darken the colour - as the oil seeps in but … it’s a great way to finish, I reckon.
Non intrusive, remove the dirt but not the patina - beautiful old board that will still look old …. just look prouder. 👍
Thank you, @BrownishGerbil very nice set! With the good advice you got from @Powderdigit you might ideally lighten it as much cleansing as you darken it oiling and waxing 
Wish you a lucky touch with it!
Thanks for the guidelines @Powderdigit, I'll have a go at the checkers side and see what the result is before I decide what to do!
Nice set!
Hi,
Today I got a very old and grimy chess set delivered and I'd like to start with giving it a good clean. Never done this before so better ask here how to go about it.
I won't touch the sides, they just show their age. The playing field however is very dirty. Seems like dust and soot have been free to accumulate during many years. It's a very thin veneer and I guess the black fields are stained or painted. There's no laquer or other finish on it.
The other side has a checkers board on which I can experiment so any tips on how to approach this without removing the black stain? Any help is much appreciated!