Proper term for people from eastern Asia

You might be able to use the term "East Asians". Of course, you have to consider whether your audience is sufficiently familiar with the East Asia region or not. Also, Mongolia is at times included as an East Asian country; and to further muddle the boundaries, some people use the term "East Asia" to also include Southeast Asia (not a practice I personally approve of).

In order to avoid any ambiguity you could define the term when you first use it in an article, e.g. East Asians (i.e. people from China, Japan, and the Koreas).

As for "Oriental", I don't believe it carries a negative connotation in most of Europe (but I'm not 100% sure on this). To me, the term carries connotations of romanticism and adventure - probably because of the Orient Express. However, it is definitely an antiquated term and on that basis alone I would recommend you not to use it. The fact that some people (possibly a large amount of people) take offence to the term is another good reason to phrase it differently. Note that both Oxford Dictionaries Online and Merriam-Webster list "Oriental" as having negative connotations. (Also note that they disagree somewhat on the geographical scope of the term, with the former dictionary interpreting it as referring in particular to East Asians and the latter as referring to Asians in general.)

And what do you mean by people from "this part of the world"? Are you referring to actual Chinese, Japanese and Korean citizens or to people who can be ethnically classified as such but are citizens of countries outside of East Asia? I can tell you that my Chinese friends (Chinese nationals) aren't even familiar with the term "oriental" and that they would refer to themselves as Chinese (and that makes perfect sense as you would most likely also refer to yourself as American rather than North American if someone asked). If they'd use a regional term they would definitely go with "Asian" - but as you already noted, this term is too broad to be of much use in many cases.

"Oriental" may not be a slur, but it is outdated, and some people may take offense because it is associated with stereotypes from old-fashioned, unenlightened times. I grew up into the 1980s calling myself "oriental," but would no sooner use it today than "Chinaman" or for that matter "colored."

In the U.S. "Asian" generally refers to East Asians and Pacific Islanders, whereas in Britain it refers to South Asians. It is true that it is inexact, but racial categories always are; thus a light-skinned Arab Egyptian-American is not an "African-American." Incidentally, "Oriental" simply means "eastern." Historically, countries like Russia, Turkey, or Persia were called oriental not inaccurately.

If you're talking about cultural heritage rather than racial identity, however, Asia does break down into broad regions: East Asia, Southeast Asia, South Asia, Central Asia, Southwest Asia. "East Asian" roughly corresponds to the predominantly China-influenced cultures of East Asia (China, Korea, Japan, Mongolia, Taiwan, Vietnam; though geographically Vietnam is of course Southeast Asia).

Outside of the USA, the term Oriental is not offensive at all.

I am Japanese and I don't really find the term "Oriental" as offensive but used in the wrong way or said in a rude way can make it pretty mean sounding. However, others may find it offensive when used under any circumstance.