The problem is simple. You can't do this easily, if at all, using Access by itself. But if you allow for a brief human intervention, you have a shot.
Don't sell him a 3-machine license. Sell him 3 single-machine licenses. We have some articles on how to program such licenses in Access. Then tell him that he has to identify the three machines on which he will use the software. You take whatever he sends as machine identification, use it to generate some sort of hash key, and you are good to go.
Here is at least one thread on this topic:
Hello, I'm totally new to trying to figure this out, I've read lots but quite confused! I've made a db that I want to add a licence key to so that the db shuts down unless a valid licence key is entered by the user. It doesn't need to be totally secure and if people go to the lengths to hack...
www.access-programmers.co.uk
NOTE: Unless you sold him the right to modify the DB, this is a case where you might wish to look at building an .ACCDE file. IF you sell this person the right to modify the DB (customize, if you prefer that language), then you cannot restrict usage at all, since in that case your customer would simply strip out the restriction code.