IMO- A tripod is not as essential as it was in "film days". When shooting with film, I'd often blow a whole roll of 36 on one subject just to ensure that I had a decent shot. I used a tripod a lot for landscapes and closeup work to ensure sharpness. It was like Christmas when I got the slides back from the lab or opened the developing tank when processing B&W film. I knew there might be a great present in there, but was not entirely sure what it would look like. With digital the results can be checked immediately, and reshot if the results are unsatifactory. Lanscapes usually hold still long enough for several shots.
On a 3-month tour we did some "on the back of an envelope" calculating to estimate how much time we spent taking pictures. Using an average of 1 minute per picture, which is very conservative, we came up with 75 hours. That is a pretty good chunk of time and equates to about 1050 miles at 14mph that we could have ridden if we chose to use the time differently. Photography has been our life-long passion, so we consider it time well spent. We also tend to concentrate on people pictures while touring, which does not lend itself to using a tripod.
Used a bridge railing for this. Sure, a tripod would have allowed me to get it "snappy" sharp, but I'm not sure that was necessary to convey place and mood. This was taken with a Canon G12, my present touring camera. I only stuck this in here becasue the Eiffel Tower was used as an example
We still use a tripod with our DSLR's, but that is when we have a need for extreme sharpness, or doing close up work.