I have jungled all but a handful of my 3s games since the Shadow Isles patch dropped, using champions from Amumu to Darius to Xin to Elise, and I will say for certain that jungling can definitely work. The stigma against it seems to come from people having a lack of experience with 3s jungling and doing a poor job, or from teammates being incapable of holding a solo lane (not as hard as it sounds) and blaming it on the jungler. Unfortunately, I've noticed that in the games where somebody doesn't believe I can jungle successfully we tend to do much worse as a team because my role isn't properly taken into account.
I tend to pick my junger based on my team's comp; that is, go tanky if we need tankiness, AP if we need AP, AD if we need that, CC corresponding to the kind we're missing, etc. In my experience, the best 3s junglers have a lot of mobility and burst plus some CC, but almost anything can work as long as it fits in with your team's comp.
Regardless of what champ you pick, I strongly suggest building one or two gp10 items ASAP and buying potions with your excess starting gold. You, more than either of your teammates, can afford to be weaker in the early game as long as you can clear the monster camps successfully, so the gp10 is absolutely worth it. I like to go p-stone on tanks or people with no sustain, Avarice Blade on any melee and Kage's on any AP. These items (minus p-stone) are more worthwhile now that they have more diverse build paths.
People laugh at Smite in TT, but I highly recommend it; you should be able to use it once per clear on the big golem in order to efficiently take down the golem camp. On your first clear, this means you should hit level 2 by the time you kill the big wraith (the second camp you should hit). It also means that if you have a fast clear time you can take out golems again before the altar first unlocks and hit level 3. I like Flash for my other spell, but that's largely a stylistic/strategic choice; if you plan on building Eleisa's on an hp-based tank then you might want to take Heal instead, for example.
What is said above about communication is important--you should encourage teammates to call mias quickly and also to be careful in the early game. Additionally, while I rarely take the sight items, keep in mind that if your enemies ever start to consistently counter-jungle you then you *must* purchase one. It's worth noting that Twin Shadows can function as a very effective sight item if used properly.
Jungling in TT takes some practice, but I find that I almost always get the highest gold on my team and often end up with the most out of both teams even if I lost--owed partly to the gp10 items and partly to the amazing gold output of the jungle. My team also gets excess levels at certain points, and while we often lose the early game we generally come back and stomp face later.
The main disadvantage to jungling is that an ADC solo bot is going to have a hard time, but I've seen some people do a great job there anyway.
Anyway. All that to say: In my experience, jungling in TT is definitely viable--you just have to know what you're doing, just like on SR.