First of all, it's likely that although you feel you have problems with your intervocalic /t/ or /d/, you probably don't. (Your intervocalic /t/ or /d/ is just the /t/ or /d/ that you make between vowel sounds) It's far more likely that you are self-conscious about your speech—and that in your search to find out why, you have honed in on the difference between how you think your /t/ or /d/ should sound and how you think your actual /t/ or /d/ sounds. This is perfectly normal.
The way that we make a sound in isolation and the way that we make it in real speech between other sounds is very different. Every sound that we make is affected by the sounds around it in complicated and fascinating ways. As native speakers, we aren't consciously aware of this fact.
In standard American English, and often in Southern Standard British English too, a /t/ will become voiced when both preceded by a vowel and followed by an unstressed vowel. When this happens the /t/ will become a tap. This means that the tongue will rapidly raise to the alveolar ridge and make a complete closure before rebounding back of the ridge. There is not enough time for the air to build up behind the closure, which is why it is a tap and not a normal /t/. Also for a majority of General American speakers, the vocal folds will keep vibrating for the duration of the consonant. This gives the /t/ a [d]-like quality.
When an intervocalic /d/ occurs between two vowels, the effect will be the same. So intervocalic /t/ and /d/ sound identical for a large number of Gen Am speakers.
When we make this sound, it is the top surface of the tip and rims of the tongue that make contact with the alveolar ridge.
If you genuinely have a slightly 'Indian'-sounding intervocalic /t/ or /d/, there are three possible causes that I can think of. The first is that you might be using the very, very tip (the apex) of your tongue to make this tap as opposed to—more or less—a normal closure like you do for a regular /t/ or /d/.
A second reason might be that you are using the underneath of your tongue to make the contact (I think this is unlikely, but this is just an opinion).
However, if your /t/ or /d/ really is Indian in quality, you may be making a retroflex contact. I'll explain what this is.
If you feel behind your top teeth with your tongue, you will feel a little flat, shelf-like bit of your mouth. Behind that your mouth suddenly arches up to form the 'roof' of your mouth. That shelf there is your alveolar ridge. We make our alveolar sounds in the middle of that ridge most of the time.
However, speakers from the Indian subcontinent often make /t, d, n, l/ just behind that ridge on the bit where the mouth is arching up. If you make your intervocalic /t/ or /d/ there, it will give you this kind of sound.
Having said all of this, if you are a native speaker, I doubt that this is a feature of you speech. More importantly, you do not need to worry about your pronunciation. How you say things, in terms of actual sounds, is not so important to good communication or relations with people. Far more important is your intonation, you body language, and what you actually are saying. If you have unusual intonation (for example if you don't have a good range of pitch when you speak), then this is worth worrying about and changing. Otherwise – enjoy your speech and use it to your best effect!
Hope this is helpful.