I also find the "designer" names annoying. But MAYBE if I had a poodle mix I MIGHT call it the -doodle name, just because those are becomming so common people would probably correct me if I didn't say it. But that would only be if I knew the mix for sure.
My dog Luna is from a shelter so we don't know where she came from, but I truely believe she's a "deer head chihuahua." They're being bred more commonly, but since I personally don't consider it a real breed, I have never told someone that that's what she is (except friends and people who know deer heads aren't a real breed). For all I know, though, she could be a chi/terrier mix or something, so sometimes I get the, "She doesn't really LOOK like a chihuahua!" I'll say she came from a shelter and is probably a mix.
We get this question a lot with the service dogs I work with, though, and it's a little more complicated there. They come from shelters so we usually don't know what they are. But telling people that the dog is a "_______ breed" is pretty much an advertisement for that breed..... so we try to be tasteful and vague. For example, we have a dog now that is truely a mix of several different things, but the best we can think to call him is an anatolian mix; but the trouble is that it's a rare anatolian that has the temperment of a service dog, especially this particular dog, so we don't necessarily want people rushing out and buying an anatolian just like the one they saw as a service dog. So to the public we just call him a mix.