It is perfectly safe and a wonderful experience. In fact, in some areas of Nepal, the only way to get around is by bike. Lumbini, for instance, the alleged site of the Buddha’s birth is actually a whole number of temples spread out over a very large area – too large to walk. A few people hire rickshaws, but they are expensive and uncomfortable. Most people hire bikes as did my partner and I. We had a great time and Lumbini holds some of our best memories of Nepal.
The country in general is very safe. We often walked around at night with no sense of threat. We’re South African, so we’re very vigilant to crime and danger, it being a big part of our lives over here. The people are peaceful and extremely welcoming to guests whom they view as a divine gift.
The only thing I would warn you about would be if you were planning to go through some of the more mountainous areas. The Nepalese are….hmmm, how should I put this…….somewhat creative and irreverant drivers who often go round sharp curves on the wrong side of the road! We were very nearly killed by a truck doing this – on two separate occassions! But there are plenty of stunning rural areas where you can see traffic coming for miles and you won’t have to worry about this.
Hope this helps :)
P.S. Sorry for the messy, hard to read answer. For some reason it’s ignoring my line breaks and mooshing the paragraphs together.