Replace the Relay
I had this exact problem, climate control blower stopped working, started working, stopped working erratically. Heat and aircon worked fine, because you can test it by pulling air through the system by cranking down a rear window and driving fast. Here are detailed directions on how to fix it, since I used this forum for ideas and found them very helpful. All information provided pertains to my 2007 3i.
As others have said, it's probably the Heater Relay. Very easy to change in the 2007 Mazda 3 (assuming this pertains to other years as well, I have no way to check). Pop the hood open, then open the fuse box on the right hand side near the battery. Take a look at the underside of the lid of the fuse box. There is a space labeled HEATER. Find this in the box. It is a black cube about an inch on each side. It does not look like a fuse. Part number is LF66 18 811. Here is all the writing on it -
LF66 18 811
When typed into the Mazda dealer's part system, this comes up as "Main Relay". The replacement part number is LF67-18-811. List price of $29.99, mine was $21.99. The dealer had to order it for me, and it took two days.
It takes less than a minute to install the part, just pop the hood open, find the box, pull the relay, push in the new one, and close everything. I fixed it for $23.42 (including tax). Don't let the dealer install it and charge you a fortune!
If you remove the relay, the blower will not work. Pop the new one in, and it works great. So - give this a try first, before you spend a thousand bucks having the blower replaced.
Thanks to all the great suggestions on this forum, you guys helped me find the answer!