Visa runs from Thailand can be done by plane, train, or bus. Note that some countries make it more difficult to enter their countries by land, and some countries require you to apply for a visa before you can enter. It might be more effort than it’s worth, unless of course you’re interested in traveling around the country. It’s worth paying attention to forums and other up-to-the-minute resources, as these details can change often.
Getting into Myanmar / Burma requires traveling by air or by land for day-trips to Thachileik (opposite Mae Sai), Myawaddy (near Mae Sot) and at Kaw Thaung / Victoria Point near Ranong. While you can cross into Burma over land, it’s really only available for a one day trip. You’re also limited as to where you can go in the country, but you can get a new Thai tourist visa when you cross back into Thailand.
Cambodia has six land crossings, and all are available to foreigners. The crossings at Poipet and Sihanoukville is quite popular, partially because they make getting around the country a bit easier. Visas are offered on arrival, and are good for 30 days. You’ll probably want to fly into Phnom Penh and the Siem Reap airports, but there are enough bus options to consider as well.
While not a standard visa run destination, it’s easier than you think. First, get to Savannakhet on the Lao-Thai border, then cross into Vietnam via the Lao Bao crossing.
Laos is surprisingly easy to enter from Nongkhai’s bus terminal, but you’ll need a tuk-tuk to get to the Friendship Bridge. This is perhaps the most common crossing, but you can also cross into Laos at Chiang Khong (for Houayxai), Thakhek (for Khammouan), and Mukdahan (for Savannakhet, but you can also get a bus from Bangkok as well). Lao visas are issued on arrival, and are good for 30 days.
Getting into Malaysia is a bit more dangerous, thanks to the various and random acts of violence in the deep south. You’ll probably be warned not to go overland by your State Department, but going through Satun is possible overland. Flying to Malaysia would be easier, but you’ll get a 30 day visa on arrival either way. As a bonus, there are Thai consulates around Kuala Lumpur, Penang, and on Singapore.
Wherever you go, ensure you pay attention to how long you can stay in the country, and how long you’ll get once back in Thailand. Arriving in Thailand by air gets you a longer visa than arriving overland.