Yes your towels will get clean in cold water.
I wash all my towels in cold water and the towels come out just as clean as they did in warm or hot water.
Actually the towels tend to last longer now that I use cold water instead of hot water and I heard that washing towels in hot water instead of cold water weakens the towels fiber, fade colors, destroy elastic and also can sometimes cause shrinkage.
So by washing your towels in the cold water you can help extend the life of your towels and you'll save money on heating the water as well.
I do a lot of my clothes washing in cold water as well unless the fabric needs hot or warm water to properly clean but cold water usually works just fine for most of my laundry including the towels.