I have long wanted to be able to make the rolled “r” sound that one finds in Italian, Spanish, Romanian, and many other languages. Until recently I was unsuccessful. But I have made major progress over the past week or so, and I thought I would share what has helped.
First, stop thinking of this letter as some kind of “r.” That can throw you off. It sometimes helps to think of it as a kind of “d” as that gets your tongue placement closer to where it should be. But better still, think of it as a completely new consonant from another language, rather than a variation on one in your own.
Second, don't flap your tongue. This sound, the alveolar trill, will occur when your tongue is in the right place, your mouth is in the right shape, and the air flow and volume are right. Just as you cannot make a “raspberry” sound with your lips by flapping them, but rather it is a sound that occurs when the air flow and lip placement is right, so too you cannot make the rolled “r” by flapping your tongue.
Third, experiment with lots of different placements of your tongue. You are going to need to hunt for the spot – and that is a crucial part of the process of learning that sometimes gets missed. Often we read that we should lift our tongue to a certain spot to make this sound, and when we do what we think we are supposed to and don't get the result we want, we assume that we are just not capable of making the sound. Don't give up – experiment!
Fourth, and most importantly: sing in Italian. While this may sound ridiculous, there is something about the air flow and other things that seem (to me, at any rate) to make it easier to achieve this trilled “r” sound when singing (and singing in full voice and not just under your breath). Start with the word forza as that was the one on which I first managed to make the sound consistently. Other combinations of “r” with vowels and consonants may be more challenging. I found that I was able to make the sound consistently when singing before I got the hang of doing it in speech (in fact, the latter is still a work in progress, but the former can serve as a stepping stone to the latter).
Here's the song that helped me:
Skip to the line with the word “forza” or just bring that line to the start for the purpose of working on this. It will probably take some additional experimentation to get other words to work, but once you do, singing “a ricordarci” is going to be fun.
Let me know if this helps you!