Early morning is the best time for travel photography.
I absolutely love to get up early and go shoot. Everything is quiet and still and you can enjoy a place without all the distraction and noise of crowds. It's just the perfect time of day for me, photography-wise.
When I travel, I always try to get up at least once so I can get out and shoot sunrise. I tend to be a morning person anyway, so it's pretty easy to do, although I will admit it can be hard to drag yourself out of bed at times. But, it is always worth it.
On our Paris trip at the end of last year, I got up really early one morning and walked over to Notre Dame. Literally, I was the only one there. I had this major tourist attraction to myself. I wandered all around and shot it from lots of different angles. It was great.
On your next trip, try doing this and see how it goes. You might need a little caffeine later, but you will be happy with the shots you get!