OP, I flew cross country twice when I was pregnant with my DD, and I've already done it 1x this pregnancy (and likely will have to again for a funeral in the near future)  and while the radiation exposure is discouraging, I don't think it really accomplishes anything to get really worried about it.  I've heard some people try to fly at night to limit exposure to the sun's radiation 
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However, if I were you, and travel for work was frequent, I would look into cutting back for other reasons. Â You said you are newly pregnant...I HOPE you don't end up with bad nausea, but you might. Â With my 1st pregnancy mine started around week 6...this time week 5. Â I actually just flew cross country at 8.5 weeks and I felt SO sick the entire time. Â The pilot barely ever took down the fasten seatbelt sign so they kept trying to make me sit down when I needed to use the bathroom to throw upÂ
 not fun. Â
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With my last pregnancy, my flights were in my 2nd and 3rd trimesters, and I was less nauseaus but I was big enough that the seats were SO uncomfortable. Â I also had to pee every 30 min which is not convenient. Â (And one of my flights was 1 day after a newsworthy airplane scare so EVERYONE was on super-alert mode and the flight attendants were VERY pushy about making people sit down)
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I can see how flying to a tropical destination might help it feel worthwhile, but if your trips are for business, I'm guessing that you will probably be less tired and much more comfortable with fewer trips! Â Congratulations on your pregnancy. Â