The short answer to this is no, Arizona 20-day preliminary notices do not need to be sent by certified mail with return receipt requested. You can find details about this and other Arizona preliminary notice requirements here. Now, here is a bit more details.

The different ways to mail notices...

I provide this just as a point of clarity. The terms "certified mail" and "certified mail return receipt requested" commonly get confused as the same thing, and the details about each are commonly confused. State laws typically "require" one of these methods.  

Generally speaking, preliminary notices can typically be sent in one of four ways:

1. Regular mail (which is just a regular stamp);
2. Certified mail (without return receipt requested). This requires a signature, and you can always access the signature, but the signature does get sent back to you. Delivery is confirmed and tracked and returned to you;
3. Certified mail with return receipt requested. This requires a signature exactly like certified mail, but the signature is returned immediately to you. Delivery is confirmed and tracked and returned to you;
4. Registered mail is an additional service you can pay for on top of these other mail types. It "registers" the mail piece and provides superior tracking of the mailpiece as it is being delivered, but is not a signature-collecting/confirming delivery, like certified mail.

See also: Construction Preliminary Notice Delivery Methods: Certified Method vs Return Receipt Requested 

Arizona requires.....confirmation of sending!

The funny thing about Arizona's preliminary notice rules is that it is pretty liberal about the type of mailing required, and instead, it puts the emphasis on sending.  

You want, in other words, to have the proof that you are sending. In Arizona, you can actually send the mail by just regular mailing without any certified or registered services, so long as you keep a "certificate of mailing"). It is common in Arizona for notices to be sent with a "acknowledgment section," so that the person receiving it will sign the acknowledgment of receipt and return it to the sender (Yeah, that happens sometimes, but it's terribly hard to rely on).

So, let me break this down more specifically.

First, the law says you can send the notice by "either first class mail with a certificate of mailing" -- which is just regular mail -- or "by registered or certified mail, with the sender obtaining a certificate of mailing, receipt of registration, or receipt of certification."

Second, "If sent properly, the notice is considered served upon mailing – not when the recipient actually receives it."

Under no circumstances do you need to send notices in Arizona with "return receipt requested." And -- in my opinion -- it's a waste of money to do so. However, because regular certified mail will make your life so much easier in Arizona (versus chasing down receipt acknowledgments, certificates of mailing, etc.), I do think it's worth paying for certified mail without return receipt.

TL;DR: Send Arizona notices by certified mail.