The Roman alphabet was indeed phonetic, more phonetic even than its closest descendent, Italian, which has lots of irregularities in its letters depending upon their placement in a word. Latin, comparatively, has few to none.
I suppose the modern German pronunciation is closest to the Latin (though there are still intervocalics in German, like Italian). Firstly, in Latin, 'c’s are always of a “hard” nature, that is, always possessing the sound at the beginning of these words: “coop, cot, cup, can, ken, cane, key,” etcetera. Therefore you might as well spell Latin words with a ‘k’ instead of a ‘c’ in English to give them their ancient pronunciation. This “hard not soft” rule is exactly the same for the letter ‘g’.
Latin vowels are also constant, and resemble their counterparts in Italian. And also like the Romance language, Latin 'r’s are trilled.
The Latin ‘s’ is constant as well, always possessing the unvoiced hiss sound (like a Greek sigma), such as “seep,” “standard,” “essay,” “sorry,” and other examples in English. French, Italian, and German, however, all voice a single ‘s’ when it comes between two vowels, making like an English ‘z’ sound; and since most of our vocabulary is based on languages like French and German, we’re used to doing this too. However, until the introduction of the Greek zeta into Latin, there was no such sound to the Romans. Thus the ‘s’ in Caesar sounds like the ‘s’ in “essive,” never like the ‘s’ in “easy.”
When I say the names of Cicero and Caeser while speaking English, unless I’m speaking with someone who’s familiar with Latin pronunciation, I usually refrain from the ancient way of saying the word, and prefer to use the standard accepted English version. That way people won’t look at me and say, “what?”
Additionally, Latin does not possess the letter ‘v’. All the Romans had was the vowel ‘u’, which, when they wrote it, looked like how we would write a V. When this vowel was very short, however, it would become semi-consonantal in nature, and resemble an English ‘w’, if a bit longer. Most modern classicists prefer to retain the ‘v’ in spelling, however, to keep Classical and Ecclesiastical spelling consistant, for one, and also to denote the consonantal ‘u’, or ‘w’ sound.
This is often the hardest aspect of Latin to come to grips with (I know it was for me), since, when one starts to pronounce all the 'v’s in Latin as 'w’s, with a bad American or English accent on top of it, like so many classicists, it sounds like hideous baby talk (and in many ways, such people likely would have sounded like lisping babies to the Romans). But once one attains a more fluid speech and begins to understand the precision of Latin phonetics (unlike our relatively careless system), it becomes one of the most beautiful languages that you ever could hear.
A counterargument made in support of the “Latin ‘v’ like the English ‘v’” defense is that the letter ‘f’ sound exists in Latin, which in English is the unvoiced counterpart to the voiced letter ‘v’. The sound of ‘f’, however, is a very strange occurrence; this sound likely did not exist in the West before its introduction to the Latins (and Germans) by the Etruscans — and the Etruscans had no voiced ‘v’ counterpart to their ‘f’ because there were no voiced consonants of any kind in Etruscan. Another argument against the counterargument is that ‘f’ is a sibilant, a whispering, hissing sound, and the other sibilant in Latin is the letter ‘s’, which, as mentioned above, had no voiced counterpart before the Greek introduction of zeta.
Long answer to a short question.
Therefore Cicero was more like “key-keh-roh,” and Caesar like “keye-sahr” (“keye” as in “eye”), rather than our Anglicized pronunciations.