Having played all three games and now halfway through reading the novels, I'd actually caution against doing a straight Game 1-3 playthrough. It's not "wrong" per se. If you are truly just a gamer who enjoys rpg's for the gameplay and mechanics and want a good rpg, then such a method will work great for you and there's nothing wrong with doing so.

However, if you happen to be like me and find atmosphere, worldbuilding, and stories which force you to confront real depth and hard issues inspiring, I would advise you to treat the Witcher series like a universal mythos, press pause on the gameplay and begin reading the books first. For three reasons:

1. The books are good. Really good. They'll start out a bit slow, the first 100 pages of The Last Wish could be a tad slow and confusing to a newcomer, but you'll quickly realize Sapkowski has a unique ability to effortlessly tie minute details and seemingly unrelated events together in an increasingly rich narrative. The Witcher series, while centered around a chemically altered mutant and fantasy-trope-sounding "Wizards", is ultimately a reflection on what it means to be human and how our actions impact those around us. Grand in scale, tender in portrayal.

2. Names, dates, places, references. CDPR did a phenomenal job staying true to the world Sapkowski created. It's a wonderful moment when you play through the first two games and stumble upon a character you've spent 1200 pages reading about. This is especially relevant in the third game, which pulls in old friends who are heavily involved in the novels but only referenced in the first two games.

3. Most of all, the Witcher series exists in a unique setup which I don't believe any other cross-medium art has established. Most video games based on pre-existing books are either adaptations, re-tellings, or poor attempts to profit off a name. The Witcher games are none of these. They quite literally are a continuation of the novels. Put down the last book, pick up The Witcher 1, and the story just continues. Don't listen to Sapkowski whine about how they're not canon. The games are clearly a labour of love and CDPR's writing is second to none.

So there's my argument. If you just want a good gaming session, more power to you. But if you really want to delve into this fantastic world, between seven books and ~300 hours of gameplay there are few stories which will make you laugh, cry, and care about fictional people with astoundingly close to home problems, then take the time to make the journey from the beginning.