For what it's worth, I would recommend against Wizard. They only have one build viable for Grifts, and would require a very rare weapon (Furnance) to be doable enough for satisfaction. Not sure about Monk - can't say I see a lot of people play that now a days. You could just go ahead and do what everyone else; create a Demon Hunter. It's the best class and I don't think Blizzard is planning on nuking it.
http://www.diabloprogress.com/rating.stat_grift_solo_s0
As you can see from the rankings (also if you click each individual class to view the Grift level), DH is doing the best, followed by Barbarian. Surely you should play what you enjoy, but D3 is kind of competitive now, so you might as well go for the best.
All classes only have one build for GRifts 35+ , ranking at the moment are like ,
1 DH
2 Barb/Wiz
3 Crusader
4 Monk/WD
Now I know one Barb as match DHs highest rift but from what I've read this was done using a conduit shrine trick (leaving a shrine and pulling a pack to it, then clearing till your close to spawning the guardian and go back to your shrine, kill pack, pop the shrine and collect your reward. )
Now saying you require a Furnace for a Wizard this is true for 40+, but below that Sunkeeper/Maximus work fine. Wizard are the least gear depended GRift clearing class to because of how Firebird works so for an ult that are quite rewarding if you like kiting gameplay.