A Messianic New Year... Well, First, When's the Jewish New Year?

There is nothing against celebrating the Gregorian, Julian, and other New Years as long as they aren't celebrated to worship pagan g-ds. Remember that nothing in of itself is unclean (cf. Romans 14). However, the Jewish (including Messianic Jewish) New Year is in Aviv/Nisan, not Tishrei. According to Nehemia Gordon, who (even though not Messianic) studies Tanakh very well:

Today few people remember the biblical name of Yom Teruah and instead it is widely known as "Rosh Hashanah" which literally means “head of the year” and hence also “New Years”. The transformation of Yom Teruah (Day of Shouting) into Rosh Hashanah (New Years) is the result of pagan Babylonian influence upon the Jewish nation...Some of the Babylonian month names found their way into the later books of the Tanakh, but they always appear alongside the Torah month names. For example, Esther 3:7 says:

“In the First Month, which is the month of Nissan, in the twelfth year of King Achashverosh.”

This verse starts off by giving the Torah name for the month (“First Month”) and then translates this month into its pagan equivalent (“which is the month of Nissan”). By the time of Esther all the Jews lived within the boundaries of the Persian Empire and the Persians had adopted the Babylonian calendar for the civil administration of their Empire. At first the Jews used these Babylonian month names alongside the Torah month names but over time the Torah month names fell into disuse.

And:

The Rabbanite Midrash relates that when God said to Moses "This month (HODESH) shall be for you the beginning of months" (Ex 12,2) the Almighty pointed up into the heavens at the crescent New Moon and said "When you see like this, sanctify! [=declare New Moon day]". This Rabbinic fairy-tale highlights an important point, namely that the Bible never comes out and says we should determine the beginning of months based on the New Moon. The reason for this is that the term for "Month" (Hodesh) itself implies that the month begins with the crescent New Moon. 

In conclusion, while I celebrate Rosh HaShanah b'Aviv, I call Yom Teru'ah "Rosh HaShanah-Kesef"-- "The Fiscal New Year". And (given that I live in America) celebrate the Gregorian New Year, but I don't worship pagan g-ds. Here's a history of the New Year, by the way:

January 1 Becomes New Year's Day

The early Roman calendar consisted of 10 months and 304 days, with each new year beginning at the vernal equinox; according to tradition, it was created by Romulus, the founder of Rome, in the eighth century B.C. A later king, Numa Pompilius, is credited with adding the months of Januarius and Februarius. Over the centuries, the calendar fell out of sync with the sun, and in 46 B.C. the emperor Julius Caesar decided to solve the problem by consulting with the most prominent astronomers and mathematicians of his time. He introduced the Julian calendar, which closely resembles the more modern Gregorian calendar that most countries around the world use today.

As part of his reform, Caesar instituted January 1 as the first day of the year, partly to honor the month’s namesake: Janus, the Roman god of beginnings, whose two faces allowed him to look back into the past and forward into the future. Romans celebrated by offering sacrifices to Janus, exchanging gifts with one another, decorating their homes with laurel branches and attending raucous parties. In medieval Europe, Christian leaders temporarily replaced January 1 as the first of the year with days carrying more religious significance, such as December 25 (the anniversary of Jesus’ birth) and March 25 (the Feast of the Annunciation); Pope Gregory XIII reestablished January 1 as New Year’s Day in 1582.