From researching the discussions and articles I've found on the subject there seems to be a consensus that the TH spelling in German was dropped at the turn of the century. Although one source seemed to place the change just after WWII (but I think that's erroneous). As NicholasW pointed out, there never was a TH sound in the German language and no one seems to know why Old German used that spelling (I'm curious for the answer!)
The only word spelled with a TH in german today is THRONE (why? I don't know, it wasn't indicated). Of course this change has affected other well-known German words such as THALER/TALER, the monetary term which gave rise to our "dollar." However, it is also indicated that many Germans have ignored this rule change and continue with the TH today...in fact, The Neanderthal Museum there still uses the "H" and also in their URL: www.neanderthal.com (although I tried this and came up with a dead page?) So while technically the H is gone its usage remains a matter of choice. I wonder if purists are requiring Germans named Thomas to become Tomas?
And, by the way, if this revision back to the hard T (or "D"...two letters interchangeable in spoken-German) would carry over into English it would be no problem for me...having grown up in North-Central New Jersey I tend to harden my TH's anyway whenever dose kinds of words dat take de change show up dere in my speech!...I know I mispronounce dem but I can't help it. (South Jersey natives have their own dialect, though, called "Piney", after the Pine Barrens...a sort of localized Southern twang).

Another question: Why do Celtic dialectics tend to harden the TH in English? i.e. Irish..."Ting" for "Thing"?