Well, I have to say Smitie covered this pretty well in her article about the Gorons.
But adressing the theory, as long as they have rocks (they probably need volcanic rocks which is probably why they only ate Dodongo's Cavern rocks in OoT) and some way to escape the water (secret passage to Death Mountain Crater in Goron City/Goron City being too high up for the water to reach) they're fine.
Also, they were probably just as likely to survive the flood as the Zoras (who became the Rito), the Kokiri (who became the Koroks), and the Hylians. After all, they could've went up to the top of Death Mountain via the secret passage in Goron City. Another possibility is that Goron City was too high up for the water to reach to begin with (seems unlikely though). The scale of Death Mountain seemed a little exaggerated but there is a chance, however slim, that Goron City was too high up for the water to reach.