As neil points out, /t/ and /d/ in English are by no means palatal stops. The palatal point of articulation is technically two points away from the alveolar ridge, maybe even three, depending on what chart you use. But the English /t/ and /d/ are perhaps the epitome of alveolar stop.
The alveolar ridge is right behind the teeth, and there's really not much a difference between the dental sound and the alveolar one. The problem is not as much place of articulation, but manner. Oftentimes non-native speakers will fail to aspirate the /t/ enough so that it sounds like our 'd', and about two different things will happen to /d/. First, it can be too voiced (an English /d/ actually sounds like a /t/ in other languages because the voicing is very lax). Then, espacially in the case of Spanish-speakers, it can end up getting pronounced as /ð/, like the 'th' in 'the', when between vowels. this is because this sound is an allophone of /d/ in some languages. And the truth is that a dental stop sounds almost exactly the same as an alveolar, and I don't believe any languages employ them as contrastive phonemes, so you could get by speaking English without being misunderstood if you pronounce them dentally.
Now, the real secret to /t/ and /d/ in English rests in a paricular allophone of both of these sounds. This allophone I speak of is the middle sound (in almost all American dialects and some British) of the words 'water' ,'butter', 'muddy', 'wader', 'waiter' and many others. This sound is the alveolar flap or tap and is the same in all of these words in fluent (uncareful) speech. This sound is basically the same as the 'r' in Spanish, Italian, Russian, Arabic, and many others. The IPA symbol for this is an 'r' without the tip on the left side, but because this medium sucks at representing IPA correctly, I will use the alternate symbol 'D'. So, these words come out as /wa:Dɚr/, /bɚDɚr/, /mɚDi/, /we:Dɚr/, etc. Phonemically, 'caddy' and 'catty' are the same word /kæ:Di/ (except in some dialects, like my on New Orleans one which make a contrast between two different /æ/ sounds, one being a bit higher up than the other). In some American dialects and the vast majority of British ones, these two are phonemically very different since the /t/ gets aspirated even in the middle of words.
Also, there are palatal stops, but they do not exist as phonemes in English. There are the actual palatal stops [c] and [
J]. Then there are the palatalized alveolar stops [tj] and [dj], which are prevalent in Russian and many of the Slavic languages. Any Russian will tell you that there is a damn big difference between palatals and alveolars. Actually, these four sounds are pronounced at roughly the same point on the roof of the mouth, but with different parts of the tongue, making the first two sound more like velars and the last two more like alveolars.