The answer is "Yes - more like traditional Judaism", but without the drama.
The only difference between traditional Judaism and Messianic Judaism is that Messianic Jews have accepted that the Messiah has already come and presented Himself as our life substitution (sacrifice) which gives us eternal life through our acceptance of His substitution or us, and our living out our lives according to the Word of YHWH (John 17:2-3). Otherwise, all the instructions of Torah given to the Jews - those instructions which still apply to us - we still need to follow and obey! Those Torah instructions include eating only clean, permitted foods.
The permitted foods are listed in Leviticus Chapter 11. So a Messianic Jew, for example, will not eat pork, shellfish, or the scavenger birds - the same as the Jew. A minor difference is that the Jew, living under "kashrut" must abide by a number of additional "rules and regulations" imposed by the rabbis. For instance, under kashrut, a piece of beef could not be purchased from a modern supermarket unless the meat had been slaughtered and prepared according to kashrut law. Otherwise, the buyer does not know how the animal was prepared, according to kashrut law, that renders the beef inedible. But the Messianic Jew realizes that the beef itself is edible (permitted) and how it was killed is not the overarching requirement of Leviticus 11. So the Messianic Jew can purchase the meat (with one exception - addressed in a moment), and, upon preparing the meat at home, remove any remaining blood by salt before cooking, and trimming any excess fat. The resulting cooked beef is completely kosher, but not kashrut.
The issue of "no eating meat with milk" is another rule of kashrut that the Messianic would not find imposed. The two scriptures on the subject say the following:
Deuteronomy 14:21 "You are not to boil a young animal in its mother's milk."
Exodus 34:26 "You are not to boil a young goat in its mother's milk."
It is from these two verses that all of Judaism forms the prohibition of eating meat with dairy. The Messianic Jew realizes that the prohibition is nonsense because having meat and dairy at the same time does not mean the meat being eaten was boiled in its mother's milk! Simply having meat and dairy at the same time in no way breaks the commandment. Take a cheeseburger for example, the chance that that cheese was made from the mother of the animal from which the meat came is almost zero. Of course one would have to be more careful if one's meat and dairy were purchased directly from the same supplier where the chance would be increased that the dairy product and the meat product came from a mother/offspring pair. But in most cases, where dairy and meats are produced by separate mass-market production facilities today, the issue is hardly worth being concerned about.
Today, with the influence of Islam in our cultures, many are worried about meats being labeled "halal". The thing is, like "kashrut" halal meat isn't actually OFFERED to the god of Islam. Halal (similar to kosher meat preparation, which is done according to Torah rules) is simply a method of preparing meat as prescribed by Muslim law via "halal butchers." Preparing meats in both religions simply centers around the method of preparing the meat; and,in neither case, is the meat "being offered" to God. However, if you actually KNOW that a certain piece of meat WAS offered to "their god," then it is, of course forbidden. (See Acts 15:29 and 1 Corinthians 10:28.)