I'm not a very flexible person naturally but have benefited a lot from yoga and basic stretches recommended all the time on here. Just two things:
(1) I agree with others that with time your body will move better and it will feel more comfortable. For me that started to really click after about a year of training. Granted, I still sucked, but I just mean I could shift my hips, spread my legs wide (that feels so weird to write), etc.
(2) I make it a point to stretch often, including during class. While my instructor is demonstrating a move I'm almost always doing some kind of stretch, even if it's just sitting with my knees folded, while still paying attention.
I have a friend who is incredibly flexible and attributes it all to rolling over the years, in that his body just adapted over time to him trying to get his knees higher and what not. I've never seen him stretch so who knows.