I also thought that I couldnt ride/train without Carbs. But I've been doing Paleo since May and am down 80 lbs (90 lbs in total since Feb). There is an entire book for Paleo for athletes that explains when & how to get the appropriate carbs prior/after big events. I rode 40 miles last saturday (19.3) average and almost 50 on sunday with a 20 - 22 mph group and the only carbs i consumed were from veggies & fruit (with the exception of one honey stinger at the mid point of both rides).
When you start a low carb diet your body goes through a process (for me it takes 2 weeks) where it adapts to burning more protien for energy. During those two weeks i do feel lathargic - but from my experience the outcome is worth the hard 2 weeks.
I currently ride 5 days a week training for a full century on March 3rd. My training rides are at a minimum of 23 miles and up to a metric century on weekends. I also swim 1 - 1.5 miles 3 days a week and lift weights for 1 hour 5 days a week. All on a low carb diet.
I personally feel like I have tried about every diet on the market - inclusive of spending 3 weeks at the Biggest Looser Resorts. One of the things I learned while at the resorts is that not all calories are equal. A 100 calorie twinkie will treat the body much worse than a 100 calorie pack of almonds.