The Spartans, like the rest of Ancient Greece, wrote and spoke in Ancient Greek which uses its own alphabet. To see that alphabet, go to Microsoft Word and change the font to "symbol". Although they probably had their own dialect, there isn't any one which we know of.
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The ancient Spartans spoke Dorian Greek, a dialect of ancient Greek language. Modern Spartans, like most other modern Greeks, speak modern Greek with the usual regional variations. ..
Using their vocal cords etc. (like most people)
Where did the ancestors of the Spartans originally come from?
Spartans were Greek, so they spoke Greek.