First assumption I had was this involved the Sabbath being kept on the multiple of 7 days of each month. As I looked it up elsewhere (the Hope For Israel site which I have many issues with starting with it's Racism) I saw that it actually involves not counting the New Moon as a work day (a detail Rob clearly missed) and making the Sabbaths the 8th, 15th, 22nd and 29th of each month.
As I contemplated this I was reminded of how I'm always puzzled by all these often conflicting Rabbinic customs about what to do if the Sabbath falls on the 14th of Nisan or 10th of Tishri or some other inconvenient day, and would ask myself "why didn't God deal with that in The Torah?"
But then as I contemplated it more I remembered that Jesus said "God made The Sabbath for Man not Man for The Sabbath". Which means you really shouldn't stress about it.
Both key arguments Rob made against it are flawed. One I alluded to already, you never work more then 6 days because of the New Moon. The other about God creating the Moon on the 4th day is that I feel God created the Moon in it's 4th day (or 18th day in a Tabernacles Creation Week theory) position, I don't think the first New Moon was a Wednesday.
The biggest flaw to me is that this theory can't explain how Pentecost works, how can the Fiftieth day also be the day after a Sabbath?
In fact First Fruits and Pentecost as a whole weaken the argument. If those days were always supposed to be the same day of a Month then God would have just said that like he did with the other Holy Days. Instead he said the morning after the Sabbath. This is my same argument against the Rabbis starting the Omer on the 16th of Nisan.
A minor observation is everything I read so far seemingly ignores that sometimes a Hebrew month has 30 days, meaning a single work day between the 29th Sabbath and the New Moon in this model. I have long theorized that in God's original perfect creation all years were 360 days, 12 months of 30 days each. I do NOT however then interpret that to mean Daniel 9 should be counted as 360 day years.
This model also overlaps with the absurd Friday Crucifixion model. The Thursday model which I favor and even more the Wednesday model need the weekly Sabbath to NOT be the 15th special Sabbath in that particular Nisan. And as Greek students know the text itself refers to there being plural Sabbaths passing between the Crucifixion and the Resurrection.
Jesus was Crucified on the 14th, Liberated Sheol on the 15th, rested on the Weekly Sabbath on the 16th, and Rose again on the 3rd day of Unleavened Bread which was the 17th of Nisan, the day Mordecai was honored and Haman hanged in Esther, and probably the day Noah's Ark landed on Ararat.
I did once did calculations and concluded that IF the original first month was Tishri as many assume, and IF my originally months were 30 days theory is true. Then the 10th and 17th of of the first Nisan would have also fallen on Sundays and the 16th would have been a weekly Sabbath. That's an interesting observation but I wouldn't read anything profound into it.
Update October 28th 2016: Lunar Sabbath supports have often sought to say the synchronized week goes back to Babylonian Paganism. But it was actually Mesopotamians who used a Lunar cycle to determine their days of Rest. As discussed (with an Anti-Judeo Christian bias) on the Wikipedia pages for Week and Sabbath. We however don't need a Sumerian origin for the word Sabbath, it comes from Sheba, the Hebrew word for Seven.