Every other semester I have a student in one of my phonetics classes who does not or cannot pronounce the [ɹ] in words such as rain, hearing, brain, tree, cry. Assuming that it's not some kind of partial tongue paralysis, which hints can be given to these students? Here is what I try to explain and demonstrate to them.
1. Try to growl like a dog. First, I demonstrate the way I want my student(s) to sound like. Here's your truly as a dog:
As a next step I try to convince her/him to imitate me (I must admit I'm usually not very successful in my attempts):
<growl> + /eɪn/, /hɪə/ + <growl> + /ɪŋ/ etc.
2. In case this demonstration does not 'convince' the student I resort to this:
2.1 Start producing a very long [dːːːːːːːːːːd]; while you do this, slowly let your tongue tip slide along the hard palate back and forth to the /d/-position: [dːːːːːːːɹːːːːːːːd]
2.2 Do this several times and gradually increase the speed of this movement.
2.3 Watch your lips in a mirror while you say <read> and <heed>. Do you round your lips when you say the /r/ in <read>? If so - good! Make sure you round your lips in the following exercises as well. If not - do round your lips while you pronounce the /r/ because lip-rounding gives this sound a more English tinge.
2.4 Now practise the r-sound intervocalically: /uːruː, ɔːrɔː, ɑːrɑː, ɜːrɜː, iːriː, ɪrɪ, ere, æræ, …/
2.5 Next, practise saying proper words with /r/ in intervocalic position as in <marry, carry, Terry, Tory, Laurie, hurry, ...>.
2.6 /r/ as a singleton in initial position is next on the agenda: <rude, run, rattle, red, ...>.
2.7 /r/ in consonant clusters is pretty much the last stage of perfection: <brought, brain, prey, trough, try, crawl, grow, ...>.
Take yourself time. After several years of wrong usage your 'bwain' is used to giving the wrong orders and it needs time to reorganise itself.