Has Or Had?

Both has and had are possible depending on the perspectives of the speaker. With a back shift persepctive, it's "had" becasue of "I didn't understand ......." which implies time passed.

DearYolandaMr. Ward has been telling us about the quiz since last week, but I didn’t understand what he has/had been saying.

... has been telling us about the quiz since last week, but I haven't understood what he has been saying.

CJ

Hi, Jim. I think "I just found out that we are having a math quiz tomorrow. " means "I" didn't understand Mr. Ward's words for a week, until just now I finally figured out what he has been talking about. Is it right?

I agree with Jim.

You have to use Haven't understood.

But if you want to use didn't understand I think Had been is the right choice because you are talking about an action that started in the past before another action in the past .so I go for past perfect continuous.

To me it sounds more natural to say I didn't understant what he had been saying.

Let's see what others tell

Hi, hrsanei. I totally trust you people, but I'm a little confused. I think "I haven't understood..." means "I didn't understand, and I still don't understand", but since "I" have known there is going to be a test, it should be "I" have understood what Mr. Ward has been saying, only that "my" understanding came a week later than that of other students...

Or is it that—it came to me just now—Jim and you think "I" knew about the test by other means (e.g. a classmate told "me" there was going to be a quiz), instead of finally understanding Mr. Ward's words?

Poor Yolanda

I just found out that we are having a math quiz tomorrow. Mr. Ward has been telling us about the quiz since last week, but I didn't understand what he has/had been saying.

DearYolanda"I just found out that we are having a math quiz tomorrow. " means "I" didn't understand Mr. Ward's words for a week, until just now I finally figured out what he has been talking about. Is it right?

No. There's nothing in that sentence that has anything to do with not understanding something for a week. That information is in the next sentence.

When you said "telling us about the quiz", I thought you meant, "instructing us how to do the math problems which would appear on the quiz", not "telling us that there would be a quiz". I answered according to what my understanding was. Apparently, that is not what you meant.

In view of the fact that you wanted to say something else, I would say it like this:

I just found out that we are having a math quiz tomorrow. Mr. Ward has been telling us that we were going to have a quiz since last week, but I didn't understand what he was saying.

CJ

Thank you, Jim! Now I see.