It is actually a kind of generalization in terms of time.
-lar/ler (plural) plus -i (possessıve suffix)
In the evenings, mornings, on Tuesdays, during the day, etc. akşamları, sabahları, salıları, gündüzleri, ...
Source: http://www.turkceciler.com/Dersnotlari/cekim_ekleri.html It is in Turkish, sorry.
Update: this seems to be the new address:
Look under "2. İYELİK EKLERİ", and also "4. -lEr ÇOĞUL EKİ".
The address has changed. Try here:
It is a long article. You can find this specific section under "2. İYELİK EKLERİ", and also under "4. -lEr ÇOĞUL EKİ".
Or you can just search for the word "sabahlar".
You're probably right. The accusative is used for "specific" direct objects, and I extrapolated that the "specific" aspect might also be applied elsewhere, for example to prepositional objects. Hopefully a native speaker will weigh in. I have seen deorme90's theory presented on another website, so the "salı günleri = salıları" theory seems like the best we've got.