Earlier I described the top secret of talking like an American. Here's another secret.
The letters 't' and 'd' do not denote the same sound in American English as they do in most Indo-European languages.
Since the time of Sanskrit (and most likely before), the 't' and 'd' have been dentals. That is, to produce the proper sounds, speakers of most Indo-European languages press the tip of the tongue against the back of the top teeth.
Not so in American English. The sounds denoted by the same characters are palatals. To produce them, one needs to press the tip of the tongue against the upper palate. The tongue does not touch the teeth at all, it is placed about one centimeter behind them. Everything else remains the same (i.e. the formation of the sound), except of course the resonation as described in The top secret of talking like an American.
The result is similar to but clearly different from the European sound. Not knowing this and continuing to use the dental sounds is one of the most distinguishing characteristics of a "foreign" accent in the US.
Immigrants often try to imitate the American pronunciation by substituting the 'd' for the 't' because they notice many Americans do that. Alas, it does not work because they continue to use the dental sound. Instead, all you need to do is move your tongue back just a bit and continue making a clear distinction between the 't' and 'd'. You'll be amazed how American you'll sound once you get used to it!