[Photo from bangkoktourist.com]
Doing a similar job to the taxi is Thailand's ubiquitous tuk-tuk (ตุ๊กตุ๊ก). So named because of the sound of their engine, these are motorized rickshaws and are popular amongst tourists for their novelty value. They are occasionally faster than taxis in heavy traffic as weaving in and out is easier, but generally about the same or slower. Without any luggage, 3 people can fit into one fairly comfortably - it's possible to fit more in but it gets a bit cramped. Fares always have to be bargained for, and it is sometimes possible to bargain tuk-tuk drivers down to less than the taxi flagfall of 35B when they make good value. Most times, they offer no savings over a taxi, except perhaps if you're good at bargaining and can speak good Thai. The initial price they quote is likely to be well over the going rate, but it's easy to bargain it down to a more reasonable one if you know roughly the equivalent taxi fare.
It's essential to bargain the price with tuk-tuks before getting in. If you only ask after the ride, it's likely to end in a request for an ridiculous fare which can obviously lead to an unpleasant situation.