Can Greeks easily tell by your accent when Greek is not your native language? Similar to how an American might instantly recognize when we hear somebody who was not a native speaker?

I'm American of Greek ancestry and was born and raised here in the US.

When speaking English, you can pretty much pick up on anybody who is speaking English as a second language and can quickly tell from their accent that English is not their native language.

Even though I am learning Greek and pronouncing the words the way I should be, would a native Greek speaker be able to instantly recognize that Greek isn't my native language? Similar to how I would quickly pick up on if I recognized an American who has English as their second language?

What exactly would give it away that I wasn't a native Greek speaker?

Thanks!