The first thing I thought when I read your OP is "what do you mean *a lot* of meat?" It's a very relative term, IMO.
If you are following Mark's advice here, you're hitting in the neighborhood of .7-1.0g protein per lb lean body weight per day, yes? If you are, let's say, 165lb male with 15% bf, that's a lean mass of about 140lbs, or in the range of 98-140g of protein per day. Let's average it, and call it 125 g of protein a day.
If you're doing some dairy, and having, let's say, a 3-egg omelet in the morning with 1 oz cheese on it, that's about 25 g of protein.
If you have a handful of almonds, that's about 8 g protein - you're up to 33g for the day
If you have a salad with salmon on it for lunch - let's be generous and say a 5oz piece - that's 30 g of protein; you're up to 63 for the day.
If dinner is a steak, and you have a moderately sized flank steak (grass-fed, of course), let's say it's 8 oz, and that's about 56g protein.
With only one not-so-big serving of red meat, you've hit almost 125g of protein for the day. You've also gotten some fish, nuts, and eggs in there.
If you do that a couple times a week, and rotate through some other proteins - pork, chicken, turkey, etc etc., the truth is you can be a very primal eater and NOT really eat a ton of (red) meat.
I know some paleo/primal folks eat a lot more meat than this (I probably do - I don't track it, so I don't know), but I also think it's not at all difficult to be on a primal diet and not consume HUGE quantities of meat.
“If you want to build a ship, don't drum up people together to collect wood and don't assign them tasks and work, but rather teach them to long for the endless immensity of the sea” -- Antoine de Saint-Exupery