I have some texts that are not religious (like dictionary) but contain The Name. And because it's not obvious that one should not recycle them, but rather put to geniza, I want to mark them somehow. So, is there some traditional phrase equivalent to 'put this to geniza' or 'contains The Names' in Hebrew? How do israelis inform about that in, let's say, religious newspapers?
The used term in newspapers is "טעון גניזה"
The Hebrew term you're looking for is "שֵׁמוֹת" - shaimos or shemot (or some other transliteration along these lines). The word literally means "names" and, in this context, refers specifically to items that contain Holy Names of God (as you've specified) or are otherwise not to be discarded due to their holiness.
Shaimos Alert - YU Pesach To-Go 5774 contains Shaimos on page 13 (and possibly other pages)
(Link added by me)
The sentence I usually see used on printed material is:
"נא לשמור על קדושת הגליון"
=Please preserve the holiness of [this printed] sheet.
I see this on material like the brochures about the weekly Torah reading which different organizations publish and distribute in synagogues.
I think that religious people are familiar with the phrasing לגניזה. That seems to cover both the content that something contains "religious materials" as well as the action itself, that you wish them buried.
BTW, there are numerous items that people assume are "shaimos" that really are not. I'll edit in a link to an article about this, later. Thus, I would caution that prior to declaring such items, make sure that this is really necessary for the items you want to "dispose". Genizah is costly - if not monetarily, it is labor and time-consuming to an extent.