I was always told as long as there are ice crystals at the center of any meat, then it would be "safe" to re-freeze, so I guess that would mean just a little after that and you should start cooking it.
After spending some time on eGullet, a food forum, I suspect there is even more leeway than that, although I suppose as a rule of thumb it is probably safe advice.
The main factors are time and tempature. If you keep them in an ice chest with ice and they stay at, say, 34F, they may be totally defrosted, yet as safe as in your refrigerator.
Perhaps it would be good to get an old mechanical thermometer to throw in there with the food.
Also, it depends on what kind of food. If I understand correctly, steaks are fairly free of critters on the inside. It's the outside that starts to grow stuff. This is why it's fairly safe to have a fairly rare steak -- the outside is cooked, killing anything there.
But if you take hamburger as an alternate example, it is ground, so the inside is the outside and thus undercooked hamburger is much less safe, reference all the e. coli associations with hamburger, esp. undercooked.
I won't go into it, but chicken should always be well cooked.
Our nose knows when meat is past due. Use it.
And of course, consider you are going to cook that puppy. 160F will kill e. coli.
Make sure you maintain proper sanitation so you don't contaminate your work surfaces and other food with the raw meats. This may take a little more attention without hot water and possibly less light.
While not exactly on topic, I asked a similar question over on eGullet that might be of interest:
eGullet: Stupid question: Meat, how long in the fridge? And getting a bit farther off topic, here is some more discussion about food safety and sanitation:
eGullet: food safety, keep the bugs awayeGullet: Washing Your Sponges, Do you do this?eGullet: Disinfecting kitchen, Nice-smelling bleach doesn't work-john