They did not have one single word like we do/Neo-Latin did with millennium. When Cicero talks about "thousands of years", he simply says multa annorum milia.
One workaround later authors used is the adjective miliarius. Tertullian (De Anima 31) has aevum miliarium to mean "a period of 1000 years." It's two words, but it does seem to be a singular unit of sorts. The only issue is that I don't see it in anything earlier than Tertullian. While it was typically used substantively in the neuter to mean "milestone marker" Augustine (Civ. Dei 20.7) uses miliarium annorum. It's quite a bit later than "Classical Rome," but it's probably the closest you'll get to "millennium."
An off-the-wall suggestion could be decem saecula, using the narrower meaning of saeculum as a period of one hundred years (and I suppose adding the decem to it would suggest that particular meaning).