Road safety on Zanzibar

Pretty safe. Of course there are occasional accidents, but you can be in the wrong place at the wrong time anywhere. If you have big bags, they will be put on the rooftop of the dala dala (public "bus"), but I haven't really heard of people having anything stolen. Keep your valuables with you.
Just note that the dala dala will be FULL and not very comfortable. Also, you will be dropped off in the village, from where it may still be a bit of a walk to the resort you have chosen. Sometimes the drivers may take you there for a little extra money, but don't count on it.
A more comfortable (and quite affordable) option are the shared minibuses, which can be boked through your accomodation. They pick up at the hotels in town early in the morning and drop off at the resorts north and east

Zanzibar is as safe as a sandpitch compared to Nairobi.

A friend of mine and I were keen to do the same trip to the East coast of Zanzibar, which might I add is absolutely stunning. Well worth it. We had thought about dala dalaing it across but they run quite infrequently compared to the mainland and wouldn't afford us the flexibility of moving up and down the mostly rural east coast by ourselves. What we decided on doing was renting motorbikes which meant we got the wind in our faces and could visit every town we fancied along the east coast right up the north of the island. We could stay in a village for as long or as little as we liked and gave a few lifts to appreciative locals. To rent a bike in Stone Town find someone offering a bike for 35,000 - 45,000 TSH for three days. We managed to get ours for 30,000 but mostly because we had quite good Swahili to bargain with, making it impossible for him to discuss behind our backs. Make sure you get a Zanzibarian permit for driving which is a little slip of paper that the bike person should organise with the local police, and if stopped [like we were by a Saddam looking policeman in an army 4x4, pull up slowly, get off the bike, take off your helmet, smile, shake his hand and show him your permit, international driving licence [£7 for a year with the AA] and make small talk. It's also useful to keep your travel insurance on you in case the policeman gets picky. A little backhander might have to take place if you've stupidly left your documents in the bag in the hotel like I did! Don't panic if he starts raving about courts and fines, he just wants some dinner money. 5,000 or 10,000 TSH should quiet him. Most of all have fun!

Hey Cobby, the motorbikes sound fun! If I was younger and not travelling with my 3 children I just might try it although, I wasn't actually as adventurous when I was younger. My husband and I did rent mopeds and rode around Cozumel once when we were travelling childless.