As an aircraft mechanic at a major airline, I think I can answer this pretty well.
It wouldn’t be hard at all. All I have to do is wait for an aircraft move. This could be to bring a plane into the hangar or just to move it from one gate to another. I could simply volunteer to taxi the aircraft, which wouldn’t be unusual at all. At my airport, we have to cross an active taxiway and runway to do this sort of thing. We do it weekly, if not daily. I’d request fuel (if needed), and instead of proceeding across the active runway, I’d just hang a right and throttle up. By the time anyone knew what was going on, I’d be airborne.
There aren’t really many precautions to prevent this. Nor are there a lot of precautions preventing me from, say, throwing a bunch of screws into an engine or chopping a wire bundle in half. The truth is that I pretty much have carte blanche access to the airplane.
What will be done? Probably nothing. The efficacy of airport security is about what you’d expect from a $12/hour job. The focus is on making sure you don’t have a large tube of toothpaste, not whether I’m going to steal an aircraft. Most mechanics have often joked about how easy it would be to “steal” an aircraft. I’m so familiar with airport procedures and policies that even if you prohibited me from doing A, B, and C, I could simply do X, Y and Z. I am a mechanic after all. My job is to think outside the box with respect to “my” aircraft as well as to think about policies and procedures we have in place. I’m probably more lawyer than mechanic. I can find a loophole in everything long before security ever will.
Furthermore, we do a really good job at policing ourselves. My coworkers and I are like brothers (and sisters). We know when someone has an alcohol problem, a marital problem, a debt problem, etc. security is just focused on the four seconds they see you. As long as you’re not acting extremely strange, and you don’t have a weapon, they’re as good as useless.
Is it possible for this to happen? Sure. But, I wouldn’t bet on it being a new way for down and out aircraft mechanics to end their lives.
EDIT: Since my original answer, the media has reported that the person in question was a ground servicing agent. I think this is an interesting twist as, where I work, most of these agents do not come from an Airframe & Powerplant background. Even the ones that do generally don't have a background on the same fleet types. They may know how to do some things in the cockpit, such as operate the fuel panel, but most of them are not familiar with all of the controls.
I suppose anybody could download a flight simulator game and learn these things, to some extent, but there's still enough going on that it would be sufficiently difficult, if not very slow, to figure out how to do things like starting an engine (which generally requires starting the APU), release the parking brake, steer with the tiller, etc...
That said, I have heard of some airlines allowing these ground service agents to tow the aircraft. That would require knowledge of starting the APU, releasing the parking brake, etc. To my knowledge, none of them are allowed to taxi, so there's still the "problem" of starting the engines. That said, it's not like starting engines is really a big deal, it's just that it's another delay in the process. I'm going to go out on a limb and take a stab that he has had some prior "higher capacity" experience with planes before this ground servicing job.
I'd also like to note that now that I think about it: when I see planes being pushed back on the ramp, their engines starting, and them taxiing out, I rarely give it a second thought. It is literally not something I would even question. You just assume the people up there are qualified, know what they're doing, and you just go about your day.