Well, yes, the "р" in Russian indeed doesn't sound exactly like the "r" in English, even if you have a rhotic accent. And yes, it is rolled, even though this "trill" part is actually very short. But there is a post-alveolar trill in the word "утро", that's for sure. Just keep in mind that it can be much shorter than you might expect.
However, in the words "смотреть" and "секретарь" the sound "p" is palatalized (in "секретарь" there are two soft "р"s, actually: the first "р" is palatalized because it comes before the letter "е", and the second one - because there is the soft sign "ь" which indicates that the previous consonant is soft), so it's another type of trill called "dental trill". Still, no "р" sound is omitted.
As for mastering this sound... I know that there are several techniques (going from "d" to "r", trilling r in "three" etc.) and I cannot say which one is better for you, but you could try watching some videos on Youtube ("Russian pronunciation: "р" sound" or something like that) - in these videos people usually try to explain the differences between the English "r" and the Russian "р" and between the hard and the soft Russian "р"s, so I think it could be useful for you. And of course, you can try more general "How to Roll your R's" videos - they could be helpful as well.
By the way, I don't think it's impossible to learn how to make this sound. It's quite similar with the Spanish "r" sound (not absolutely the same, but still very much alike), and there are many English speakers coping with it, so I'm sure you can achieve this trill with enough practice. Good luck!
Josh Kruger, where did you heard them, could you share a link? Perhaps it is just peculiarity of someone's pronunciation....Anyway:
- Russians do roll r's
- there is 'hard r' and 'soft r' (palatalized r).
"Soft r" can be articulated as a "trill" or as a "tap"... ( English has such a "tap" in some pronunciations of 'gotta':-)).
1) why do смотреть and секретарь have different R's? Or was that just a typo? I assume they both have 'soft' r.
2) (just curiousity) where did you read about Russian soft r described as a "dental trill"? A honest question, I'm just curious about sourses for Russian phonetics around.