It's amazing what a big problem foreigners have with the "ы" sound! And at the same time many linguists believe that this sound doesn't even make a distinct phoneme! And they have their reasons. One evidence is there is no single word starting with the letter "ы" in Russian. Another is that there are such words as "предыстория" (prehistory) or "разыменование" (dereferencing) which have roots "история" and "имя", i.e. starting with "и", but after the hard consonants of the prefixes this "и" turns into "ы". The same thing happens on junctions of words, for example if I say the phrase "мой друг и товарищ" quickly, without a stop, the conjunction "и" will sound like "ы".
All these reasons mean that "ы" is only an allophone, i.e. a variant of the "и" phoneme. It is a product of the assimilation of the "и" vowel with the preceding hard consonant.
Therefore the problem of the sound "ы" is not the problem of that sound, it's the problem of the correct pronunciation of hard consonants. A person studying Russian should pay attention to mastering the difference between soft and hard consonants. And once he succeeds the "ы" problem will disappear all by itself, because Russian hard consonants are so hard that it is impossible to pronounce the "и" sound after them.