When I was in college, I had been studying Russian for TWO F*CKING YEARS without really understanding the difference between hard and soft consonants -- so, for example, if I saw -ля- in a word, I would pronounce it something like -лъйа-.

But a few weeks into my third year of Russian, I suddenly grokked (©1961 Роберт Хайнлайн) the distinction between ла and ля, and so my "American English accent" was immediately reduced, almost overnight. It was like that scene in My Fair Lady, where Eliza instantly and magically switches from the Cockney "The rine in Spine styes minely in the pline" to "The rain in Spain stays mainly in the plain"!

Of course, I still spoke Russian with a foreign accent, but after that epiphany in my 3rd-year class, my accent became much less "American."

But there are disadvantages to mastering (or nearly mastering) the "native accent", if you're a foreigner. When I lived in Moscow (93-94), the women in the grocery stores were almost always VERY rude and nasty to me -- and I think the problem was that they mistook me for a native Russian who was either mildly retarded or drunk, because I had a very "clean" accent, yet I often had trouble finding the right word, and spoke rather slowly, and sometimes stuttered.

But my American friend Brian -- who knew maybe 20 words of Russian -- could walk into a store and say "Я кочу купит клэб. Йэст у фас КЛЭБ, поджалста?"... and immediately the saleswomen would be politely showing him their selection of bread, or offering to take him to the БУЛОЧНАЯ store just up the street.

So, there are times when it's better to have a "heavy" English accent.