Merhaba! I have been studying recently with a book by now-deceased Princeton University professor Lewis V. Thomas that was originally published in 1967 and then updated in 1986. (It was a gift.)

In discussing the future tense, he says this: "Verb stems which end in e or a change the final e to i and the final a to ı before the immediately following (buffer) y of the future suffixes."

So, this would make it anlıyacağım.  "I will understand."

But here at the site, I learned: anlayacağım.

Is he teaching some fine point that has fallen out of usage?

I also wonder if the "i" in diyecek and yiyecek are holdovers from this earlier usage.

Or maybe I am completely misunderstanding this.

I don´t know how it was at that time for the written turkish but since 1988´s when I went to school, it has not been like this. The right is "anlayacağım". You get the word stem by taking off the -mak/-mek and you add -cağım, -ceğim. 

But one thing is for sure that when speaking, you never say "anlayacağım", you just say "anlıcam".

but there are some exceptional verbs like "yemek"  and "demek", they become "yiyeceğim" and "diyeceğim".

anyone who wants to add something?

 Yes I have finally realised not to say anlayacağim etc : and now I am starting to understand more what Turkish people are saying ,because I was always listening for acağım eceğim, and from now on I will not use it when speaking and maybe they will start to understand me too:

I think that they´re saying it like: anlicam(I will understand). beklicem(I will wait),etc...

Aynen