It IS possible to get a transit visa through Saudi for a bike. It IS NOT possible to pick one up at a border.
The standard transit visa form is for cars, planes and ships. This throws a lot of Saudi embassy staff because people applying for motorcycle transit visas are extremely rare. There is a form and it is possible. Do not argue or get angry with the staff, just explain politely that there is a form that must be sent from Riyadh for people travelling on motorcycles.
You need to have a visa for the countries on either side of Saudi - UAE and Jordan are easiest.
The Saudis will want you to provide a FULL itinerary - ie every major road, every major town, every overnight, all the hotels you plan to stay at, the people you intend to meet etc. The more you give them, the more trusting they will be. You can fight and rail against this all you like but without it you will not get a visa. Once you get there you can, of course, do whatever you like.
You need a sponsor in Saudi to write a letter of invitation to you. There are two ways of doing this: if you have a business, are a professional, or have some other reason to justify a work-type visit, write to someone and ask them to invite you. I have known of students to contact universities, doctors to contact hospitals, engineers to contact companies from their own country etc - use your imagination. I also recall people on this site offering to sponsor people. The other, and preferred method, is to get an invitation from the Chamber of Commerce (Saudi embassies will often raise this option). This is not as hard as it sounds - if you can justify a reason to meet with them send a letter and request a meeting. The Chamber is a much more dynamic institution than the govt.
Most important of all, give yourself heaps of time to sort out the visa. Have several copies of every piece of paperwork you think might be relevant. Each time you make contact with the embassy, give them another stack. Also, give them a few passport photos each time(worthwhile getting dozens of these things). I had four contacts with the embassy last time I applied for a visa.
This all sounds like hard work, but in Saudi you will get to see places that very few tourists get to.
Before getting too uptight at the Saudis, remember they have to deal with a massive influx of Asian pilgrims each year. They also take their role as keepers of Islam's holiest sites very seriously.