I don't know any Korean. However, I have read about it, and the thing that you mention hearing is listed as a feature of Korean pronunciation in A Reference Grammar of Korean, by Samuel E. Martin, page 31.
Martin says that ㄷ+ㄱ that is originally across two syllables ends up being turned into ᄁ, a tense consonant at the start of one syllable (and that /t/ is likewise lost before other tense consonants, including ones that became tense because of coming after a consonant, or before aspirated consonants). However, I think that this is an optional part of fast speech, so that you could also hear a [t] sound before it sometimes.
Actually, something a bit similar is supposed to happen in English: supposedly, /t/, /d/ and /n/ in English can assimilate in place to a following consonant, making "good book" sound like "goob book", "bad cold" sound like "bag cold", and "ten piles" sound like "tem piles". But this is not something that English speakers generally notice or think is happening (and I've read that even if it is hard to hear, the tongue position might still include the original position of /t/, /d/ and /n/), so it wouldn't necessarily be recommended for an English learner to make a special effort to try to pronounce things this way. But it can help to know this kind of thing so that you don't get confused when listening.